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AGRICULTURE 


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UNIVERSITY  OF  ILLINOIS 

Agricultural  Experiment  Station 


BULLETIN  No.  288 


PERSISTENCY  OF  LACTATION 
IN  DAIRY  COWS 

A  Preliminary  Study  of  Certain  Guernsey 
and  Holstein  Records 


By  W.  L.  GAINES 


URBANA,  ILLINOIS,  APRIL,  1927 


CONTENTS 

PAGE 

INTRODUCTION. .  355 

THE  PROBLEM 356 

THE  METHOD 357 

RESULTS  FROM  GUERNSEY  RECORDS 367 

Age  and  Persistency 368 

Age  and  Initial  Rate  of  Yield 369 

Age  and  Yield 370 

Persistency,  Initial  Rate,  Age,  and  Yield 371 

Correction  Factors 375 

Influence  of  Season 376 

Variability  in  Persistency 377 

Variability  in  Initial  Rate  of  Yield 380 

Increasing  Rate  of  Yield  with  Advance  in  Lactation 381 

Rate  of  Yield  and  Yield  for  the  Year 382 

Influence  of  Heredity  and  Environment 392 

RESULTS  FROM  HOLSTEIN  RECORDS 397 

Age  and  Persistency , 397 

Age  and  Initial  Rate 398 

Persistency  and  Initial  Rate 398 

Correction  Factors 402 

Variability  in  Persistency 402 

Variability  in  Initial  Rate  of  Yield 402 

Rate  of  Yield  and  Yield  for  the  Year 403 

Influence  of  Heredity  and  Environment 405 

DISCUSSION. 408 

Selection  of  Records 408 

Chemical  Interpretation  of  the  Lactation  Curve 409 

Breed  Lactation  Curves 411 

Measures  of  Persistency 414 

Correction  Factors  for  Length  of  Record 414 

Persistency  as  a  Heritable  Character 415 

The  Short-Time  Test 416 

SUMMARY  AND  CONCLUSIONS 422 

LITERATURE  CITED..  .424 


PERSISTENCY  OF  LACTATION 
IN  DAIRY  COWS 

A  Preliminary  Study  of  Certain  Guernsey 
and  Holstein  Records* 

By  W.  L.  GAINES,  Chief  in  Milk  Production 

INTRODUCTION 

The  term  persistency  of  lactation  is  used  to  refer  to  the  degree  with 
which  the  rate  of  milk  secretion  is  maintained  as  lactation  advances. 
Cows  ordinarily  reach  their  highest  rate  of  milk  secretion  in  any  one 
lactation  period,  shortly  after  calving.  Following  the  flush  of  lactation, 
the  rate  of  milk  secretion  declines  more  or  less  rapidly  until  the  cow 
goes  dry  naturally  or  is  dried  up  artificially  by  discontinuing  milking. 

It  is  clear  that,  other  things  being  equal,  the  more  persistent  a  cow 
is  (that  is,  the  less  rapidly  she  declines  in  rate  of  milk  secretion),  the 
more  milk  she  will  produce  in  a  year 's  time.  It  is  commonly  stated  that 
the  dairy  breeds  produce  more  milk  per  year  than  the  beef  breeds  or 
unimproved  cattle  partly  because  they  are  more  persistent  milkers.  As 
between  the  cows  of  the  dairy  breeds  it  is  recognized,  in  turn,  that  some 
are  more  persistent  than  others.  In  the  official  testing  of  the  dairy  breeds, 
the  short-time  test,  such  as  the  7-day  test,  has  been  very  adversely 
criticised  because  it  does  not  depend  upon  and  does  not  measure  (it  is 
said)  the  quality  of  persistency.  In  general  persistency  of  lactation  is 
counted  as  a  very  important  factor  in  the  yearly  yield  and  economical 
production  of  milk.  As  a  matter  of  fact,  however,  our  knowledge  of 
persistency  of  lactation  as  a  character  of  dairy  cows  is  very  limited  so 
far  as  exact  quantitative  analysis  is  concerned. 

The  American  Guernsey  Cattle  Club  has  published  about  15,000 
yearly  records  of  Guernsey  cows  in  detail  by  calendar  months.  The 
Holstein-Friesian  Association  of  America  has  published  about  1,500 
7-day  records  made  at  least  eight  months  after  calving,  together  with  in 
each  case  a  7-day  record  made  early  in  the  same  lactation.  These  breed 
records  afford  data  of  great  value  for  quantitative  study  of  persistency, 
and  the  present  paper  is  the  outcome  of  an  attempt  to  analyze  certain 
of  them  from  the  standpoint  of  the  persistency  of  lactation  shown  by 
the  individual  records. 


•Submitted  for  publication  June  29,  1926. 

355 


356  BULLETIN  No.  288  [April, 

The  Guernsey  records  used  (1,676)  were  limited  to  original  entries 
published  in  Vols.  33  and  34  and  No.  1  of  Vol.  35  of  the  Herd  Register. 
Only  365-day  records  which  represented  a  single  lactation  and  in  which 
conception  did  not  recur  within  6  months  after  calving  were  used.  This 
selection  was  intended  to  eliminate  any  disturbing  influence  of  pregnancy 
on  persistency.  Also,  only  those  records  were  used  in  which  not  more 
than  75  days  elapsed  from  calving  to  the  middle  of  the  first  full  calendar 
month  of  the  record,  in  order  that  the  records  should  be  fairly  com- 
parable with  respect  to  the  time  after  calving  at  which  they  started. 

The  Holstein  records  used  (1,395)  constitute  all  the  eight-months- 
after-calving  records  published  in  Vols.  24  to  31  of  the  Advanced  Regis- 
ter Year  Book,  with  certain  few  exceptions  noted  later.  The  volumes 
named  include  nearly  all  the  records  of  this  class  which  have  been 
published. 

THE  PROBLEM 

The  problem  with  respect  to  the  Guernsey  records  may  be  pre- 
sented by  considering  the  two  records  reproduced  in  Fig.  1.  Mere  in- 
spection of  these  records  is  sufficient  to  show  that  one  of  the  cows 
(10233)  was  a  very  persistent  milker.  Ignoring  the  record  for  June  be- 
cause it  is  for  only  part  of  the  month,  it  is  apparent  that  there  is  only  a 
slight  decrease  in  the  monthly  milk  yields  with  advance  in  lactation. 
The  corresponding  fat  yields  show  a  tendency  to  increase  rather  than  to 
decrease.  The  other  cow  (10372)  shows  by  her  record  a  tendency  to 
decrease  very  rapidly  in  monthly,  milk  and  fat  yield  with  advance  in 
lactation.  She  would  be  classed  as  a  very  non-persistent  milker. 

It  is  not  sufficient  for  quantitative  study  to  say  that  one  cow  is 
persistent  and  another  is  not.  The  problem,  first  of  all,  is  to  derive  a 
numerical  value  for  persistency  as  shown  by  the  individual  record.  This 
in  itself  is  somewhat  difficult  and  offers  opportunity  for  difference  of 
opinion  as  to  what  may  be  accepted  as  the  best  value  for  the  purpose. 

The  two  records  given  in  Fig.  1  represent  opposite  and  rather  ex- 
treme cases  of  persistency.  What  we  should  like  to  have  is  a  completely 
representative  picture  of  the  breed  as  a  whole  with  respect  to  the  per- 
sistency character.  Having  given  an  acceptable  numerical  value  of 
persistency  for  each  record  of  a  suitable  group  of  cows,  such  a  representa- 
tive picture  may  be  obtained  by  the  usual  statistical  methods.  The 
character  may  then  be  studied  after  the  same  fashion  as  milk  yield,  fat 
yield,  or  fat  percentage  have  already  been  studied  by  various  investi- 
gators, notably  at  the  Maine  Station  (cf.  Gowen8). 

Some  of  the  questions  relating  to  persistency  are:  the  form  of  its 
frequency  distribution  curve;  its  relation  to  the  age  of  the  cow;  its 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


357 


relation  to  yearly  yield;  its  relation  to  the  rate  of  yield  at  the  flush  of 
lactation;  its  relation  to  genetic  factors;  its  relation  to  environmental 
factors.  In  short,  what  are  the  attributes  of  persistency  of  lactation  as  a 
characteristic  of  dairy  cows  as  shown  by  the  advanced  registry  records? 


10233  CHOICE    OF    ELMWOOD    51908 

Official  Year's  Record,  Class  A 

Milk 

Butt 

er  Fat 

Sire  Sir  Ponto  of  Elm  wood  25492. 
Sire  Katonah's  Mack  17252. 

1920 
June  7, 

Iba. 
919.2 

% 
4.28 

Ibs. 
39.34 

Dam  Faithful  of  Elm  wood  39350. 

July. 

1204.5 

4.34 

51.88 

Dam  Flossie  of  Elmwood  35949. 

Aug., 

1181.9 

4.30 

51.53 

Sire  Rex  of  Eastside  12763  A.  H. 

Sept.. 

1179.6 

4.40 

51.90 

Dam  Dolly  Vardftn  of  Eagtside  20756. 
Breeder  M.  E.  Gifford,  Sherman.  N.  Y. 

Oct., 

Nov., 

1  230  .  1 
1181.0 

4.60 
4.47 

50.86 

52.82 

Owner  George  S.  Love.  Waukesha,  Wig. 

Dec., 

1212.0 

4.51 

54.06 

Born  Dec.  9,  1913.     Coined  June  4,  1920. 

Jan.,  1921 

120.">.4 

4.75 

57  .  2(i 

Sened  Apr.  4,  1921. 

Feb.. 

1077.8 

5.05 

54.43 

Requirement  for  admission-.  360.00  Ibs.  fat. 
Supervised  by  Wisconsin  Station. 
3  milkings  daily. 

Mar.. 
Apr., 
May. 

1115.  8 
!  109.1 
1113.5 

4.81 
4.78 
5.45 

63.67 

53.01 
60.09 

June  6, 

223.8 

5.45 

12.20 

Total 

14820  3 

4   68 

653.25 

1WZ  DORA   OF   ELMENDORF   56322 

Official  Year's  Record,  Class  A 

Milk 

Buttrr  Fat 

Sire  Imp.  Lord  Mar  V.  1S901  A.  H. 

191  ft 

Ibs. 

% 

Ibs. 

Sire  Imp.  Lord  Mar  14359  A.  R. 

Dec.  22. 

407.3 

4  .56 

22.67 

Dam  Imp.  Count***  I.  of  Los  Nour-ttr-s  36184. 

Jan.,  lf»20 

1009.4 

4.56 

73.39 

Dam  Imp.  Dora  of  the  Vranguo  VI.  361!).'!. 

Feb., 

12-14.1 

4.96 

01.71 

Sire  Imp.  Galaxy's  Sequel  16004  A.  R. 

Mar., 

1144.8 

4.55 

52.09 

Dam  Dora  of  the  Yranguc  R.G.A.S.  5572  P.S. 

Apr., 

1022.9 

4  .  52 

46.24 

Breeder  J.  B.  Haggin,  Lexington.  Ky. 

May. 

892.0 

4  .  98 

44.42 

Owner  R.  M.  Cooper,  Jr..  Y\  wacky,  H.  C. 

June, 

0*i6.8 

4  .  57 

30.03 

Born  Nov.  15.  1913.     Calcetf.  I>r.  1C,  l'Jl':> 

July. 

427.2 

1  .  50 

19.48 

Serwl  Oct.  8.  1020. 

Aug., 

310.9 

5.19 

16.60 

Requirement  for  admission:  300.00  ll>*.  fut. 

Sept.. 

:«>9  s 

1  ,  59 

14.22 

Supervised  by  South  Carolina  Station. 

Oct.. 

Sss  .  '> 

5  .  40 

15.60 

3  milkings  daily, 

Nov., 

223,7 

4  .  67 

10.45 

Hec.  20. 

I.V)  9 

1  .  <>3 

7.22 

Total 

8792   « 

4  71 

414   11 

FIG.  1. — PHOTOGRAPHIC  REPRODUCTION  OF  Two  RECORDS  FROM  VOL.  35 

OF  THE  GUERNSEY  HERD  REGISTER 

These  two  records  are  chosen  as  striking  examples  of  the  differences 
in  persistency  shown  by  the  published  records.  They  serve  also  to  illus- 
trate the  various  data  given  in  connection  with  each  record.  The  leaves  of 
the  Herd  Register  were  removed  and  backed  by  sheets  of  gummed  paper 
board  and  then  cut  into  cards,  one  to  each  individual  record.  Derived 
data  for  each  record  were  recorded  on  its  card.  The  records  were  thus 
brought  into  convenient  form  for  manipulation  for  statistical  purposes. 


As  indicated,  the  first  step  is  to  derive  a  numerical  expression  for 
persistency,  and  a  method  of  doing  this  we  may  consider  in  connection 
with  the  two  records  given  in  Fig.  1  as  examples. 


THE  METHOD 

The  expression  of  persistency  of  lactation  used  in  the  present  treat- 
ment is  essentially  an  old  one.   Sturtevant18  studying  the  average  yield 


358  BULLETIN  No.  288  [April, 

of  a  herd  of  cows,  used  the  method  of  expressing  the  decrease  in  yield 
of  milk  from  month  to  month  as  a  percentage  of  the  yield  of  the  previous 
month,  and  found  that  as  thus  expressed  the  decrease  tended  to  be  con- 
stant. This  expression  or  its  complement,  the  expression  of  the  milk 
yield  for  any  month  as  a  percentage  of  the  yield  of  the  preceding  month, 
has  been  used  by  various  investigators,  and  gives  a  numerical  measure 
of  persistency,  the  meaning  of  which  is  readily  grasped. 

Brody  et  all  have  put  the  expression  into  the  closely  related  form  of 
an  exponential  equation.  While  this  form  may  not  be  quite  so  easily 
understood  as  Sturtevant's  it  is  on  the  whole  a  more  reasonable  ex- 
pression, giving  concisely  both  the  rate  of  decrease  and  the  rate  of  yield. 
From  experience6  with  the  use  of  the  exponential  expression  and  its 
application  to  the  group  performance  shown  by  Guernsey  records,  it 
seemed  to  the  writer  that  it  might  be  applied  to  individual  records  to 
give  a  reasonable  numerical  expression  of  persistency. 

In  theory  the  idea  is  that  the  rate  of  milk  secretion  is  continuously 
decreasing  with  advance  in  lactation  in  accordance  with  the  equation: 


i/ 

in  which  y  =  yield  in  pounds;  t  =  time  in  months  from  calving;  -|- 

at 

is  the  rate  of  yield  in  pounds  per  month;  e  (=  2.71828)  is  the  base  of 
natural  logarithms;  a  is  the  theoretical  initial  rate  of  yield;  and  k  is  the 
rate  of  change  per  month  in  the  rate  of  yielda.  In  this  equation  k  is  the 
factor  which  is  used  as  a  measure  of  persistency.  The  minus  sign  means 
that  the  rate  of  yield  is  decreasing  and  k  taken  as  a  positive  value  is  a 
measure  of  this  decrease,  differing  from  Sturtevant's  expression  of  the 
percentage  decrease  per  month  in  that  it  is  reckoned  as  occuring  con- 
tinuously instead  of  at  monthly  intervals.  The  rate  of  decrease  is 
thus  referred  to  the  rate  of  yield  at  the  immediate  time  rather  than  to 
the  rate  of  yield  a  month  preceding. 

We  may  consider  the  application  of  equation  (1)  to  the  records  of 
Fig.  1.  In  the  first  place  it  is  evident  that  the  records  are  based  on  the 
calendar  month,  which  varies  from  28  to  31  days,  and  it  is  necessary  to 
make  allowance  for  this  fact.  This  is  conveniently  done  by  reducing 
the  monthly  yields  to  an  average  yield  per  day. 

It  is  also  apparent  from  Fig.  1  that  there  is  a  tendency  for  the 
percentage  of  fat  in  the  milk  to  increase  from  month  to  month  with 
advance  in  lactation.  This  is  characteristic  of  the  effect  of  advance  in 


"The  mathematical  relations  involved  in  the  derivation  and  use  of  the  equation 
are  given  in  more  detail  in  Bulletin  272". 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows  359 

lactation,  as  has  been  shown  in  considerable  detail  by  Turner.19  Altho 
it  happens  in  the  two  records  shown  in  Fig.  1  that  the  average  fat  per- 
centage is  very  similar,  as  between  a  larger  number  of  individuals  there 
inevitably  will  be  marked  differences  in  this  particular  which  should  be 
taken  into  account.  These  individual  and  monthly  differences  in  fat 
percentage  and  correlated  percentage  of  other  solids  are  taken  into 
account  by  dealing  with  estimated  energy  yields  in  terms  of  4-percent 
milk.  The  estimation  is  made  by  the  formula5  F.C.M.  =  AM  +  15^, 
where  F.C.M.  is  "fat-corrected  milk,"  M  is  milk;  and  F  is  fat,  all  in 
pounds.  One  pound  F.C.M.  =  one  pound  4-percent  milk  =  331  large 
calories. 

Table  1  gives  the  average  daily  F.C.M.  values  for  the  eleven  full 
calendar-month  records  of  the  two  cows  in  Fig.  1.  The  record  in  this  form 
shows  more  clearly  than  does  Fig.  1  the  marked  difference  in  persistency 
of  the  two  cows.  The  values  are  given  graphically  in  Fig.  2.  The  next 
step  in  the  work  is  to  apply  equation  (1)  to  the  data  of  Table  1.  This 
cannot  be  done  directly;  the  equation  in  form  for  application  may  be 
written, 

yd  =  Ae-k<  (2) 

in  which  yd  is  the  yield  for  a  month  expressed  in  terms  of  pounds  of 
F.C.M.  per  day.  Time,  t,  is  reckoned  to  the  middle  of  the  month  and 

365A          e'5k  —  e~5k 
—=-  =  'a  —    —  j—  —  .    Equation  (2)  may  be  converted  to  a  linear 

I  —  K 

form  by  taking  logarithms  on  both  sides,  giving, 

logic?/  d  =  logio  A  —  kt  Iogi0e  (3) 

By  the  use  of  equation  (3),  values  for  A  and  k  are  readily  derived 
from  the  observed  values  of  Table  1.  Using  the  method  of  least  squares, 
the  equation  for  No.  10233  becomes  yd  =  41.854  «««««,  and  for  No. 
10372,  yd  =  74.525  e~  202893'.  The  corresponding  curves  are  given  in 

di/ 
Fig.  2.  If  we  wish  to  use  equation  (1)  for  cow  No.  10372  we  have  --  = 


X  74.401e  -*°a»3«,  that  is,  a  =         X  .998344.     In  the  case  of  No. 

I  _  I  — 

10233  the  A  constant  is  affected  only  slightly  in  the  third  decimal  by  the 
same  transformation.  The  k  constant  is  not  affected  in  passing  to  the  form 
of  equation  (1).  For  practical  purposes,  therefore,  in  the  present  data,  we 

365  .  dy       ,  365  ^ 

may  assume—  X  yd  =  -jr  and  —  X  A  =  a. 

We  have  thus  expressed  the  rate  of  yield  or  lactation  curve  in  terms 
of  two  constants,  A  and  k,  the  values  of  which  have  been  determined. 


360 


BULLETIN  No.  288 


lApril, 


TABLE  1. — AVERAGE  DAILY  F.C.M.  YIELDS  OF  Two  Cows  IN 

GUERNSEY  ADVANCED  REGISTER 
(Records  for  11  full  calendar  months) 


Full  calendar 
month 

Average  daily  F.C.M. 
yield  (pounds) 

No.  10233 

No.  10372 

First  

42.9 
40.2 
41.7 
43.5 
42.2 
42.1 
43.3 
44.6 
40.4 
41.3 
43.7 

56.3 
49.1 
40.0 
36.8 
33.0 
23.8 
14.9 
12.2 
11.2 
11.3 
8.2 

Second  

Third  

Fourth  

Fifth  

Sixth  

Seventh  

Eighth  

Ninth  

Tenth  

Eleventh  

70 
?60 

^50 

•£ 

tJ 

b-1 
^0 
Si 

_J 
1 

•§30 
F 
"5 

>°20 

\ 

N 
N 

0.10 
0.10 

255 
572 

•    < 
o     < 

• 
e 

\ 

\ 

\ 

\ 

\ 

• 

\ 

• 

• 

w 

• 

,    \, 

• 

• 

• 

V 

\ 

\ 

\ 

\ 

\ 

\ 

b1-0 
DC 

10 

0 

\ 

X 

^ 

f"v-_J 

^ 

3                2                4                (,                &               10              12 

Months  after  Calving  (t) 
FIG.  2. — EXTREMES  IN  PERSISTENCY  (TABLE  1)  AND  FITTED  CURVES 

Equations  of  curves,  yd  =  Ae~kt:  No.  10233,  yd  =  41.9e'002<;  No.  10372,  yd  = 
74.5e~203'.  Similar  curves  have  been  derived,  one  for  each  of  the  1,534  Guernsey 
records  and  1,395  Holstein  records  studied.  The  individual  curves  are  studied  with 
reference  to  the  A  and  k  constants,  particularly  the  k  constant,  which  is  used  as  a 
measure  of  persistency. 


1927]  PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows  361 

While  it  is  apparent  from  inspection  of  the  records  as  first  given  in  Fig.  1 
that  there  is  a  great  difference  in  the  persistency  of  the  two  cows,  we  have 
now  a  numerical  basis  for  the  comparison  of  this  difference  in  the  factor 
k.  No.  10372  is  given  a  value  of  .203  and  No.  10233,  a  value  of  -.002. 
Similar  values  may  be  determined  for  other  cows  from  their  records. 

It  is  convenient  in  the  recording  and  in  the  subsequent  handling 
of  the  values  of  k  to  multiply  by  1000.  This  product,  k  X  103,  may  be 
considered  as  expressing  the  rate  of  decrease  in  per  imTe  per  month. 

In  the  case  of  No.  10233  the  rate  of  energy  yield  with  advance  in 
lactation  is  not  decreasing  but  increasing.  Since  we  are  treating  k  as  a 
positive  value  representing  the  rate  of  decrease,  we  in  this  case  must  give 
the  value  of  A;  a  minus  sign.  It  would  be  absurd  to  suppose  that  this 
increasing  rate  of  yield  would  continue  indefinitely,  especially  in  ac- 
cordance with  the  equation,  but  the  fact  of  the  record  is  that  the  rate  of 
yield  is  increasing  and  this  is  simply  recognized  as  a  negative  decrease. 
We  shall  see  later  that  an  appreciable  proportion  of  the  records  studied 
show  this  same  feature. 

While  we  are  concerned  primarily  with  persistency  of  lactation, 
that  is,  the  k  constant,  we  have  to  consider  also  the  significance  of  the  A 
constant.  By  the  equation,  t  =  0  at  calving,  and  accordingly  at  that 
time  Ae~kt  =  A;  that  is,  the  initial  rate  of  yield  is  the  value  of  A.  It 
will  be  apparent  that  A  is  thus  a  hypothetical  quantity  since  the  full 
rate  of  yield  is  not  realized  immediately  at  calving.  In  the  case  of  No. 
10233  the  rate  of  yield,  A,  is  actually  realized  at  a  time  later  than  calving. 
In  the  case  of  No.  10372  the  rate  of  yield,  A,  is  never  realized.  No.  10372 
is,  of  cour  e,  an  extreme  case.  In  general,  the  larger  the  value  of  k  the 
more  the  value  of  A  exceeds  the  maximum  realized  rate  of  yield.a  But 
nevertheless  it  will  be  apparent  that  A  is  closely  related  to  the  maximum 
realized  rate  of  milk  secretion. 

Figs.  4  and  5  are  intended  to  illustrate  further  the  properties  of 
the  A  and  k  constants  of  the  equation. 

The  determination  of  the  A  and  k  constants  has  been  carried  out  by 
the  use  of  graphic  methods  illustrated  and  described  in  Fig.  3.  The  alge- 
braic and  geometric  processes  illustrated  in  Fig.  3  are  all  straightforward 
but  there  is  an  element  of  weakness,  from  the  standpoint  of  mathematical 
precision,  in  fitting  the  straight  line  by  the  eye.  In  many  cases  the  plotted 
observations  are  quite  irregular  and  the  fitting  of  the  straight  line  re- 
quires the  exercise  of  judgment.  The  determinations  of  A  and  k  lack 
absolute  precision  on  this  account.  Some  records  were  so  irregular 
that  it  was  felt  that  extraneous  factors  must  be  so  affecting  the  lactation 

"This  fault  of  the  expression,  if  it  is  a  fault,  might  easily  be  remedied  by  dealing 
with  the  lactation  curve  starting  some  time  after  calving,  for  example,  by  taking  the 
time  origin  at  one  month  after  calving. 


362 


BULLETIN  No.  288 


[April, 


curve  that  for  the  purpose  of  the  present  study  more  trustworthy  re- 
sults would  be  secured  by  not  including  such  records.  On  this  ground, 
out  of  1,676  records  considered,  142,  or  8.5  percent,  were  excluded. 

In  fitting  the  straight  line  it  was  attempted  to  make  its  slope 
correspond  to  the  tendency  of  the  plotted  observations;  and  to  adjust 
its  level  to  the  average  level  of  the  observations.  As  a  check  on  the  ac- 


Spring  Clamp -/' 
k  Scale  - 


o    .oi    .02   .03   .6U   .05    .06    .07 


Origin  Scale 


Mo.  9498 
A  -  52.1 
k  «.05S 


Rate  of  Yield  fy, )  Log  Scale 


FIG.  3. — ILLUSTRATING  THE  METHOD  OF  DETERMINING  THE  A  AND  k 
CONSTANTS  OF  THE  LACTATION  CURVE 

The  equation  (2)  of  the  curve  is  yd  =  Ae~kl,  in  which  yd  is  the  yield  for  a  month 
expressed  in  terms  of  F.C.M.  per  day;  and  t  is  time  in  months  from  calving  as  origin 
and  reckoned  to  the  middle  of  the  month. 

Reference  to  Fig.  1  shows  that  there  are  11  full  calendar  months  in  the  cow's 
record  and  it  is  to  these  11  observed  values  that  the  curve  for  each  cow  has  to  be 
fitted.  The  record  (see  Fig.  1)  gives  directly  the  pounds  of  milk  and  the  percentage 
of  fat  for  each  month.  The  number  of  days  in  the  month  varies  from  28  to  31.  The 
average  daily  F.C.M.  yield  is  computed  by  the  use  of  a  500  mm.  slide  rule  provided 
with  a  specially  graduated  slide.  Consider  a  30-day  month  and  let  M  =  recorded 
milk  yield  (Ibs.)  and  /  the  corresponding  fat  percentage,  then  the  observed  yd  = 

~30~~  =  M  (-01333  +  .005/).    A  unity  graduation  (marked  30)  is  made  on 

the  slide,  and  with  this  coinciding  with  the  unity  graduation  on  scale  D  of  the  rule 
a  graduation  (marked  2.0)  for  2.0  percent  fat  is  made  on  the  slide  opposite  2333 


1927]  PERSISTENCY  OP  LACTATION  IN  DAIRY  Cows  363 

( =  .01333  +  2  x  .005)  of  scale  D.  Similarly  fat-percentage  graduations  are  made  by 
intervals  of  .1  to  9.0  percent. 

If  the  milk  yield  for  a  30-day  month  is  1,071  pounds  and  the  fat  percentage  3.33, 
the  slide  is  set  as  in  the  illustration  with  the  "30  unity"  opposite  1071  on  scale  D,  and 
the  runner  is  set  at  3.33  on  the  fat-percentage  scale  of  the  slide.  Under  the  runner 
the  value  321  is  read  on  scale  D,  and  the  observed  yd  =  32.1.  To  take  care  of  months 
of  28,  29,  or  31  days,  it  is  clear  that  it  is  only  necessary  to  provide  appropriate  unity 
graduations  on  the  slide. 

Equation  (2)  is  transformed  to  the  linear  logarithmic  expression: 

logio  yd  =  logio  A  -  .4343fa  (3) 

In  fitting  equation  (3)  graphically  we  have  to  plot  yd  on  a  log  scale  and  time 
on  an  arithmetic  scale.  (The  effect  of  the  variation  in  the  length  of  the  calendar 
months  is  negligible  here.)  For  the  purpose  of  this  plotting  there  are  mounted  on  a 
drawing  board  two  parallel  guide  bars,  spaced  to  accommodate  the  length  of  the  slide 
rule  between  them.  These  bars  are  each  provided  with  11  equally  spaced  notches  in 
opposite  pairs.  The  slide  rule  is  provided  at  either  end  with  a  small  lug  to  engage 
these  notches.  The  arithmetic  time  spacing  is  thus  provided  for.  The  logarithmic  yd 
spacing  is  provided  for  by  the  construction  of  the  slide  rule  itself. 

The  time  origin  (calving)  is,  of  course,  variable  with  respect  to  the  month,  being 
anywhere  from  15  to  75  days  preceding  the  middle  of  the  first  full  calendar  month. 
To  locate  this  origin  a  scale  is  fixed  alongside  the  extension  of  the  guide  bars,  gradu- 
ated in  days,  30.5  graduations  equaling  one  space  on  the  guide  bars.  Zero  of  this 
scale  corresponds  in  position  to  the  upper  edge  of  the  rule  when  the  latter  is  placed 
in  notch  1,  for  the  first  full  calendar  month. 

The  description  may  be  completed  by  following  thru  a  determination.  In  oper- 
ation the  plotting  paper,  of  plain  white  letter  size,  is  properly  placed  according  to  the 
level  of  production  of  the  record  under  consideration,  and  held  in  position  by  the 
spring  clamp  shown  at  the  top  of  the  illustration.  The  rule  is  placed  in  notch  1  and 
the  line  ff  is  drawn  along  its  upper  edge.  The  value  of  yd  for  the  first  full  calendar 
month  is  computed  as  above  outlined  and  plotted  by  a  small  circle  made  thru  the 
eye  of  the  pointer  on  the  runner.  Graphically  this  is  logio  yd,  unity  of  the  scale  D 
representing  10  pounds.  (It  is  not  necessary  actually  to  read  the  value  of  yd.) 

The  rule  is  then  moved  to  notch  2  and  yd  for  the  second  month  plotted;  and  so 
on  to  the  eleventh  month.  With  the  rule  still  in  notch  11  the  line  gg  is  drawn  along 
its  upper  edge. 

The  rule  is  then  removed,  and  with  a  celluloid  triangle  used  as  a  straight  edge  the 
line  hh  is  drawn  to  fit  the  plotted  values  by  inspection.  The  line  ii  is  then  drawn  at 
right  angles  to  gg  thru  the  point  of  intersection  of  lines  gg  and  hh.  This  is  accom- 
plished by  adjusting  the  triangle  against  a  guide  strip  (not  shown)  at  the  bottom  of 
the  apparatus. 

The  rule  is  then  adjusted  on  the  time  origin  scale  according  to  the  number  of 
days  from  calving  to  the  middle  of  the  first  full  month  (32  days  in  the  example), 
and  the  corresponding  line  jj  drawn.  Without  moving  the  rule  the  runner  is  so  set 
that  the  eye  of  the  pointer  coincides  with  the  intersection  of  lines  jj  and  hh.  The 
corresponding  reading  on  scale  D  gives  the  value  of  A,  32.1  in  the  example. 

The  value  of  k  is  proportional  to  the  distance  between  the  points  of  intersec- 
tion of  lines  hh  and  ii  with  line  ff.  This  distance  corresponds  to  10  units  of  time  and 
therefore  represents  4.343/c  in  terms  of  the  spacing  of  scale  D  of  the  slide  rule.  This 
scale  is  based  on  500  mm.  between  the  unity  graduations.  Accordingly  a  distance 
of  2171  mm.  (4.343  X  500)  corresponds  to  a  value  of  k  =  1  and  217.1  mm.  corre- 
sponds to  a  value  of  k  =  .1.  A  suitable  k  scale  is  easily  prepared  being  simply  a  uni- 
formly graduated  decimal  scale  with  a  length  of  217.1  mm.  between  the  zero  and  the 
.  1  graduations.  The  value  of  k  is  obtained  by  placing  the  k  scale  alongside  of  line  ff 
with  the  zero  on  the  line  ii,  and  reading  the  scale  at  the  point  of  intersection  of  lines 
ff  and  hh.  In  the  example  given  k  =  .058. 

This  completes  the  solution  of  the  particular  record.  The  device  takes  care  of 
values  of  yd  from  10  to  100.  In  the  few  cases  where  yd  values  of  less  than  10  are  en- 
countered, they  are  handled  by  multiplying  the  milk  yields  by  a  suitable  factor  and 
correcting  the  reading  for  A  accordingly.  Where  unusually  large  values  of  k  are  in- 
volved, as  in  the  case  of  No.  10372,  Fig.  1,  the  necessary  plotting  range  is  secured  by 
the  use  of  two  regular  sheets  properly  adjusted. 

Only  the  heavier  solid  lines  are  drawn,  in  practice,  the  lighter  broken  lines  being 
shown  for  the  sake  of  explanation. 


364 


BULLETIN  No.  288 


[April, 


FIG.  4. — FORM  OF  THE  THEORETICAL  LACTATION  CURVE  WITH  k  CONSTANT 
(=   05)  AND  A  VARIABLE 

Cows  represented  by  these 
curves  are  given  the  same  per- 
sistency value,  and  it  will  be 
clear  that  this  does  not  mean 
they  have  the  same  rate  of 
absolute  decrease  in  rate  of 
yield.  If  the  ordinates  of 
these  curves  were  plotted  on 
a  logarithmic  scale,  the  curves 

Q  ^>^        I -py^  ^^  would     appear    as    straight 

parallel  lines,  that  is,  all  with 
the  same  slope.  Constant 
persistency  then  means  that 
the  rate  of  decrease  bears  a 
constant  ratio  to  the  rate  of 
yield;  or  the  slope  of  the  lac- 
tation curve  expressed  in 
logarithms  is  constant.  It 
may  be  noted  that  the  areas 
under  the  curves  are  pro- 
portional to  the  values  of 
A,  or  any  other  ordinate. 
Curves  of  the  same  per- 
sistency result  in  yearly 

4  a  12          yields  proportional  to   their 

Time  -  Months  A's. 


§40 

fc 


u 


2 

t 


FIG.  5. — FORM  OF  THE  THEORETICAL  LACTATION  CURVE  WITH  A 
CONSTANT  (=  40)  AND  k  VARIABLE 


\ 


Time  -  Months 


If  the  ordinates  of  these  curves 
were  plotted  on  a  logarithmic  scale, 
the  curves  would  appear  as  straight 
lines  and  with  slopes  proportional 
to  their  k's.  The  measure  of  per- 
sistency fc,  is  therefore  based 
directly  on  the  slope  of  the  lactation 
curve  when  expressed  in  terms  of 
logarithms.  The  areas  under  these 
curves  would  be  inversely  propor- 
tional to  their  k's  (where  k  >  0) 
if  the  time  considered  were  greatly 
extended.  For  the  arbitrary  period 
of  a  year  the  relation  between  k 
and  yield  is  not  simple,  the  yield 

1  —  e~12fc 
being    proportional   to        — > 

K 

with  A  constant.    However,  within 
the  range  of  the  most  frequent  values 


of  k,  equal  changes  in  k  result  in  approximately  equal  inverse  changes  in  yield. 


curacy  of  the  fitting,  a  number  of  records  (205)  were  selected  at  random 
and  the  theoretical  twelve  months'  yields  computed  from  the  cor- 
responding equations.  In  determining  the  theoretical  yield  due  regard 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


365 


was  had  for  the  time  after  calving  at  which  the  record  started  in  or- 
der that  the  computed  yield  should  be  comparable  with  the  ob- 
served yield.  The  yield  as  thus  computed  was  divided  by  the 
observed  (F.C.M.)  yield.  The  distribution  of  the  resulting  ratios  is 
shown  in  Fig.  6.  The  mean  ratio  is  1.0001,  indicating  a  very  good  aver- 
age agreement.  The  standard  deviation  of  the  ratios  is  .0194,  and 
accordingly  the  computed  yield  equals  the  observed  yield  ±1.3  percent 
of  the  observed  yield.  It  appears,  therefore,  that  the  curves  have  been 
determined  with  a  very  fair  degree  of  accuracy,  as  judged  by  the  areas. 
(See  also  Table  17.) 


/n 

equency 

r  s  i 

— 

ii_ 

n 

r__r_T 

-Gb= 

34     56     .98     1.00    102    1.04    1.06 

Ratio 

FIG.    6. — SHOWING   THE    RATIOS    OF   COMPUTED 
YIELDS  TO  ACTUAL  YIELDS 

The  data  give  an  indication  of  the  accuracy 
of  the  estimation  of  the  constants  of  the  lactation 
curve  to  the  observed  values.  See  also  Table  17. 


The  Holstein  records  studied  consist,  for  each  cow,  of  two  7-day 
records  in  the  same  lactation,  the  first  of  which  was  started  not  less  than 
six  days  after  calving  and  the  second  not  less  than  eight  months  after 
calving.  They  afford  thus  only  two  observations  for  the  determination 
of  the  lactation  curve.  In  some  cases  more  than  one  7-day  record 
eight  months  after  calving  is  reported,  and  in  such  cases  only  the  first 
in  point  of  time  is  here  considered. 

The  same  curve  as  above  described  has  been  fitted  algebraically 
to  these  two  observations.  Obviously  this  does  not  give  as  satisfactory  a 
basis  as  the  eleven  observations  of  the  Guernsey  records,  but  it  is  all  the 
published  data  afford.  Certain  features  of  the  derivation  of  the  lactation 


366 


BULLETIN  No.  288 


[April, 


curve  from  the  Holstein  records  are  pointed  out  in  connection  with  Fig.  7. 

The  yields  are  considered,  as  in  the  Guernsey  data,  on  an  energy 

basis  in  terms  of  F.C.M.  The  constants  of  the  lactation  curve  are  com- 


puted  thus:  k  = 


loge  yl  -  loge 


-;  and   A    =   logr1  (loge  yt  + 


where  y\  =  yield  for  the  week  of  the  first  test  in  pounds  F.C.M. ;  y2  = 
yield  for  the  week  of  the  second  test  in  pounds  F.C.M.;  ti  =  time  in 


1 


o 

I 

cc 


Time>» — *- 

FIG.  7. — ILLUSTRATING  CERTAIN  CONDITIONS  IN  THE  LACTATION  CURVES 
DERIVED  FROM  HOLSTEIN  RECORDS 

The  solid  line  represents  the  curve  of  equation  (1).  The  dotted  lines  represent 
deviations  from  (1)  associated  with  the  preceding  and  concurrent  pregnancies  (cf. 
Fig.  246).  If  the  tests  are  conducted  at  1  and  2  we  may  expect  normal  relations; 
if  at  1  and  4,  too  high  values  for  A;  and  A ;  if  at  3  and  2,  too  low  values  for  k  and  A ; 
if  at  3  and  4,  too  low  a  value  for  A.  The  Holstein  records  give  no  information  as  to 
occurrence  of  conception.  The  influence  of  pregnancy  illustrated  at  the  right  of  the 
diagram  does  not  become  appreciable  before  the  fifth  or  sixth  month  of  gestation. 
The  effect  on  the  lactation  curve  has  been  avoided  in  the  Guernsey  records  by  deal- 
ing only  with  those  records  in  which  gestation  was  not  far  enough  advanced  to  be  a 
material  factor. 


months  from  calving  to  middle  of  first  test;  and  fe  =  time  in  months  from 
calving  to  middle  of  second  test.  As  in  the  Guernsey  data,  k  expresses 
the  rate  of  decrease  per  month  in  the  rate  of  yield ;  but  the  rate  of  yield 
is  expressed  in  pounds  F.C.M.  per  week.  The  value  of  A  in  the  Hol- 
stein lactation  curves  has  therefore  to  be  divided  by  7  in  order  to  be 
directly  comparable  with  A  in  the  Guernsey  lactation  curves.  A  table 
of  natural  logarithms  and  a  20-inch  slide  rule  have  been  used  freely  in 
the  computations. 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


367 


Records  in  which  more  than  139  days  elapsed  from  calving  to  the 
start  of  the  first  test  were  excluded ;  also  records  were  excluded  in  which 
less  than  130  days  elapsed  between  the  start  of  the  two  tests.  Nineteen 


E55 


35 


55  75  95 

Pays  to  First  Test 


135 


155 


FIG.  8. — TIME  OF  CONDUCTING  HOLSTEIN  RECORDS 

The  columns  show  the  distribution  of  the  records  with  respect  to  the  time  from 
calving  to  start  of  the  first  test.  The  curve  shows  the  corresponding  mean  time 
between  the  two  tests.  The  correlation  between  the  two  times  is  r  =  —.498  ±  .014. 
The  mean  time  after  calving  for  the  second  test  is  about  the  same  (around  270  days) 
regardless  of  the  time  of  conducting  the  first  test. 


records,  or  1.3  percent,  were  thus  excluded.    The  distribution  of  the 
first  tests  with  respect  to  the  time  after  calving  is  given  in  Fig.  8. 

Some  further  details  of  the  methods  used  are  given  in  connection 
with  the  presentation  of  the  results. 


RESULTS  FROM  GUERNSEY  RECORDS 

Perhaps  one  of  the  first  points  of  interest  in  connection  with  persis- 
tency values  is  the  form  of  the  distribution  curve  of  these  values  for  the 
breed  as  a  whole.  It  develops,  however,  that  it  is  necessary  to  make 
certain  corrections  to  the  values,  and  the  several  relations  involved  in 
these  corrections  will  be  considered  first. 


368 


BULLETIN  No.  288 


[April, 


Age  and  Persistency. — It  is  well  known  that  milk  yield  is  greatly 
affected  by  the  age  of  the  cow.  It  has  been  shown  by  Sanders16  and  by 
Gaines  and  Davidson6  that  persistency  varies  also  with  the  age  of  the 
cow,  older  cows  tending  to  be  less  persistent  than  younger  cows.  The 
mean  persistency  values  for  various  age  classes  of  the  Guernsey  records 
under  study  are  given  in  Table  2  and  shown  graphically  in  Fig.  9. 

TABLE  2. — VARIOUS  AGE  CLASSES  AND  CORRESPONDING  MEAN  VALUES  FOR 

PERSISTENCY,  THEORETICAL  INITIAL  RATE  OF  YIELD,  AND  YIELD 

FOR  THE  YEAR:  GUERNSEY  RECORDS 


Mean  values 

Age  in  years 
(class  mid-points) 

Number 
of  records 

kX  103 

Persistency 

A 
Initial  rate 
of  yield 
(pounds 
F.C.M.  per 
day) 

Yield  for 
year  (pounds 
F.C.M.) 

1.25.. 

2 

35.0 

24.0 

7  000 

1.75  

46 

36.7 

29.8 

8  674 

2.25  

410 

32.2 

33.1 

10  020 

2.75  

202 

33  6 

35  2 

10  559 

3.25  

137 

44.0 

39.4 

11  202 

3.75  

130 

46.0 

39.7 

11  092 

4.25  ... 

117 

51  2 

43  3 

11  756 

4.75  

70 

59.3 

45.1 

11  657 

5.25  

81 

56.7 

45.9 

12  043 

5.75  

51 

47  2 

42.9 

11  833 

6.25  

62 

62  4 

46.8 

11  871 

6.75  

32 

60  6 

44.8 

11  688 

7.5..  

76 

53  7 

46  8 

12  500 

8.5  

44 

55  7 

47  5 

12  454 

9.5  

38 

62  1 

46  1 

11  974 

10.5...      . 

16 

66  9 

45  3 

11  188 

11.5  

10 

62  0 

43  6 

11  100 

12.5  

4 

52  5 

47.5 

12  750 

13.5  

3 

35  0 

34  0 

10  500 

14.5  

2 

75  0 

49  0 

11  000 

15.5  

0 

16.5  

1 

65.0 

40.6 

9  500 

It  will  be  observed  that  the  relation  of  the  k  values  to  age,  while 
somewhat  irregular,  is  certainly  not  linear.  The  Maine  investigators 
have  used  an  equation  of  the  general  form,  y  =  a  +  bx  +  ex2  +  d  log  x, 
to  express  the  relation  between  age  and  yield,  and  the  precedent  of  their 
usage  is  followed  in  choosing  an  equation  to  describe  the  age-persistency 
data.  The  equation  is  a  purely  empirical  one,  and  no  particular  biological 
significance  can  be  attached  to  any  of  its  constants.  As  a  mathematical 
description,  however,  it  seems  well  adapted  to  the  data. 

The  equation  has  been  fitted  to  the  age-persistency  data  by  the 
"star-point  method"  of  Smith.17  This  is  essentially  a  "cut  and  try" 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


369 


graphic  method.  The  smooth  curve  thus  derived  has  a  close  resemblance 
to  the  usual  relation  found  between  age  and  yield.  Further  information 
as  to  the  relation  between  persistency  and  age  is  afforded  by  consider- 
ation of  the  several  interrelations  of  the  variables  listed  in  Table  2. 


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Age  -Years 

FIG.  9. — ILLUSTRATING  THE  CHANGE  IN  PERSISTENCY  WITH  ADVANCING  AGE, 

GUERNSEY  RECORDS 

Equation  of  the  curve:  y  =  21.8  +  3.00z  -  .2206z2  +  31.78  log  x,  x  being 
the  age  in  years.  According  to  the  equation  y  reaches  a  maximum  at  x  =  9.945  = 
9  years,  11  months,  10  days,  at  which  time  y  =  61.5.  That  is,  the  rate  of  decrease 
in  the  rate  of  yield  per  month  at  that  age  is  61.5  per  mille  per  month. 

In  the  equation  y  —  a  +  bx  +  ex2  +  d  Iogi0z,  the  value  of  x  at  which  y  reaches 
either  a  maximum  or  a  minimum  is  obtained  by  differentiating  y  with  respect  to  x, 

setting  the  first  derivative  equal  to  zero,  and  solving  for  x.  This  gives ^  =  b  +  2  ex  + 

(1         \     / 1  \  CJ^C 
—  1  ( -  )  -  0.  and  2cxz  +  bx  +  .4343d  =  0.  from  which  we  obtain 
loge  10/  \x/ 

-b  ±  (b*  -  3.4744 cd)* 

x  =  -. • 

4c 


Age  and  Initial  Rate  of  Yield. — The  mean  initial  rates  of  yield  for 
several  age  classes  are  given  in  Table  2  and  shown  graphically  in  Fig.  10. 
The  smooth  curve  is  of  the  same  type  as  above  and  has  been  fitted  by  the 
method  of  moments  as  developed  by  Miner14  with  some  final  arbitrary 
adjustments  in  the  constants.  Apparently  the  relation  between  age  and 
theoretical  initial  rate  of  yield  is  more  regular  than  is  the  relation  be- 
tween age  and  persistency.  A  general  parallelism  between  the  two 
curves,  Figs.  9  and  10,  is  evident. 


370 


BULLETIN  No.  288 


[April, 


Age  and  Yield. — The  change  in  yield  with  age  has  been  extensively 
studied.  Since  we  are  dealing  here  with  a  comparatively  small  popula- 
tion, the  age-yield  data  afford  something  of  a  check  on  the  representa- 
tiveness of  the  selected  records.  The  numerical  data  are  given  in  Table 
2,  the  graphic  presentation  in  Fig.  11. 

It  will  be  noticed  that  all  three  of  the  age  curves  are  more  or  less 
similar,  indicating  that  there  is  a  positive  correlation  between  A  and 
A;  of  the  lactation  curve  equation.  The  method  of  partial  correlation 
would  give  an  insight  into  the  relation  between  any  two  of  the  variables 


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Age  -Years 

FIG.  10. — SHOWING  THE  CHANGE  IN  INITIAL  RATE  OF  YIELD  WITH  ADVANCING  AGE, 

GUERNSEY  RECORDS 

Equation  of  the  curve:  y  =  10.56  -  .534x  -  .0132z2  +  40.317  log  x,  where  x 
is  age  in  units  of  6  months  with  origin  at  1.5  months.  This  gives  a  maximum  for 
y  at  x  =  17.554  =  8  years,  10  months,  25  days  of  age.  At  this  age  the  rate  of  yield 
is  47.3  pounds  of  F.C.M.  per  day  by  the  equation. 


with  other  variables  constant,  if  the  regressions  were  linear.  Plainly 
the  condition  of  linear  regression  is  not  satisfied  over  the  entire  age 
range.  From  the  data  as  plotted  it  is  evident  that  linear  regression  may 
be  approximated  by  dealing  with  ages  under  five  years.  These  ages 
include  1,114  records,  or  72.6  percent  of  the  total,  and  are  used  in  the 
correlation  treatment  following. 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


371 


Persistency,  Initial  Rate,  Age,  and  Yield. — The  means,  standard 
deviations,  and  coefficients  of  variability  of  the  variables  mentioned  are 
given  in  Table  3.  All  the  possible  simple  and  partial  correlations  are 
given  in  Table  4.  The  simple  correlations  have  been  derived  by  the  usual 
method,  using  class  intervals  of  10  in  k  X  103;  2,  in  A;  six  months,  in 
age;  and  1,000  pounds,  in  F.C.M.  yield.  The  partial  correlation  coeffic- 
ients have  been  derived  by  the  general  formulae, 

J.  fy.  \/      J.        \  J,  (~,  \y"      «  "\ 

7*12.3  =  i- .  „   -i,,   and  r  12.34  =    r-  ^   .  -i,/  • 


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IS 


17 


FIG.  11. — ILLUSTRATING  THE  CHANGE  IN  YIELD  WITH  ADVANCING  AGE, 
GUERNSEY  RECORDS 

Equation  of  the  curve:  y  =  2393.8  -  434.14x  +  .2842x2  +  13951.4  log  x,  x 
being  the  age  in  units  of  &  months.  Accordingly  y  reaches  a  maximum  at  x  =  14.129 
=  7  years,  24  days,  at  which  age  the  yield  for  the  year  is  12,362  pounds  F.C.M. 


The  simple  correlations  of  Table  4  show  that  there  is  a  significant 
correlation  between  any  two  of  the  variables.  The  closest  relationship 
is  evident  between  the  initial  rate  of  yield  and  the  yield  for  the  year. 
The  initial  rate  of  yield  is  also  closely  related  to  age  and  to  persistency. 
The  relation  of  persistency  to  age  and  to  yield  for  the  year  is  less  close 
than  the  other  relations  noted. 


372 


BULLETIN  No.  288 


[April, 


TABLE  3. — STATISTICAL  CONSTANTS  OF  THE  VARIABLES  INDICATED  FOR 
AGES  UNDER  FIVE  YEARS:    GUERNSEY  RECORDS 


Constants 

k  X  103 
Persistency 

A 

Initial  rate 
(pounds  F.C.M. 
per  day) 

Age  at 
calving 
(years) 

Yield  for 
year 
(pounds  F.C.M.) 

Mean  

39.5  +     .6 

36.70  +  .17 

2.98  +  .02 

10612  +    41 

Standard 
deviation  

30.2  +     .4 

8.63  +  .12 

.85  +  .01 

2307  +    29 

Coefficient  of 
variability  .... 

76.44  ±  1.09 

23.51  ±  .34 

28.40  ±  .25 

21.74  ±  .28 

NOTE. — The  probable  errors  of  the  coefficients  of  variability  thruout  this  paper 

are  computed  by  the  approximate  formula,   ECT  =    — -  — . 

M 


TABLE  4. — SIMPLE  AND  PARTIAL  COEFFICIENTS  OF  CORRELATION  BETWEEN  PER- 
SISTENCY, INITIAL  RATE  OF  YIELD,  AGE,  AND  YIELD  FOR  THE  YEAR: 
GUERNSEY  RECORDS,  UNDER  FIVE  YEARS 


Variables  correlated 

Variables  constant 

Coefficients 

Persistency  and  initial  rate  

.480  +   016 

Persistency  and  age  

.268  +  .019 

Persistency  and  yield  

—  .267  +  .019 

Initial  rate  and  age  

.490  +  .015 

Initial  rate  and  yield  

.691  +  .011 

Age  and  yield  

.308  +  .019 

Persistency  and  initial  rate  

Age  
Yield  

.415  +  .017 
953  +   002 

Age  and  yield  

.935  +  .003 

Persistency  and  age  

Initial  rate  

042  +   020 

Yield  

.382  +  .017 

Initial  rate  and  yield  

.009  +  .020 

Persistency  and  yield  

Initial  rate. 

—  943  +   002 

Age  

—  381  +  .017 

Initial  rate  and  age  

-.933  +  .003 

Initial  rate  and  age  

Persistency  .   .  . 

428  4-  017 

Yield  

403  +   017 

Persistency  and  yield  

.141  +  .020 

Initial  rate  and  yield  

Persistency  .  . 

968  +   001 

Age  

645  +   012 

Persistency  and  age  

.955  +  .002 

Age  and  yield  

Persistency 

409  +   017 

Initial  rate  

—  048  +   020 

Persistency  and  initial  rate  .  . 

-.025  ±  .020 

1927]  PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows  373 

The  question  arises,  to  what  extent  is  persistency  related  to  age  with 
initial  rate  of  yield  constant,  and  to  initial  rate  of  yield  with  age  con- 
stant? The  partial  correlation  coefficients  give  an  index  to  these  re- 
lations. Persistency  is  closely  related  to  initial  rate  of  yield  with  age 
constant,  the  partial  correlation  being  .415  +  .017.  On  the  other  hand, 
with  initial  rate  of  yield  constant,  persistency  appears  to  be  independ- 
ent of  age,  as  shown  by  the  coefficient  .042  ±  .020.  This  relationship  is 
significant  in  considering  the  proper  correction  of  the  persistency  values. 

Some  of  the  partial  correlations  of  Table  4  have  no  particular 
meaning  except  as  a  rough  check  on  the  computations.  For  example, 
if  yield  is  constant,  persistency  and  initial  rate  of  yield  bear  to  each 
other  a  definite  relation  and  we  should  expect  a  correlation  of  1.  The 
correlation  found  is  .953.  Two  reasons  are  apparent  for  the  failure  to 
realize  the  perfect  correlation;  first,  the  regressions  are  not  exactly 
linear;  and,  second,  the  persistency  and  initial  rate  values  of  the  lacta- 
tion curves  have  not  been  perfectly  adjusted  to  the  observed  yield.  In 
connection  with  the  latter  reason  see  Fig.  6.  The  low  partial  correlation 
between  persistency  and  age,  with  initial  rate  of  yield  and  total  yield 
constant,  is  to  be  expected  since  the  constant  factors  automatically  re- 
quire that  persistency  also  be  constant,  while  age  remains  variable. 
Similar  comment  might  be  offered  for  others  of  the  partial  correlations, 
but  in  general  it  may  be  said  that  they  show  a  fair  degree  of  consistency 
in  the  data. 

Correction  Factors. — The  evidence  on  the  relations  between  persis- 
tency and  age,  and  between  persistency  and  initial  rate  of  yield,  show 
that  a  correction  factor  for  persistency  should  be  based  on  the  initial 
rate  of  yield  rather  than  on  age.  The  correlations  indicate  quite  clearly 
that  such  relation  as  exists  between  persistency  and  age  is  associated 
with  the  relation  between  age  and  initial  rate  of  yield.  The  correlations, 
of  course,  say  nothing  as  to  whether  persistency  is  due  to  initial  rate, 
or  whether  initial  rate  is  due  to  persistency.  It  seems  reasonable  to  pre- 
sume that  persistency  is  distinctly  affected  by  the  initial  rate  of  yield, 
and  that  for  purposes  of  comparison  correction  of  the  persistency  values 
should  be  made  accordingly. 

We  shall  return  to  the  full  number  of  records  in  deriving  the  per- 
sistency correction  factors.  The  correlation  surface  is  given  in  Table  5. 
The  constants  from  the  correlation  surface  are  given  in  Table  6.  The 
regression  equation  derived  from  the  constants  of  Table  6  is  k  X  10s  = 
1.782  A  —  25.59.  The  curve  of  this  equation  and  the  mean  persistency 
values  of  the  several  initial  rate  classes  are  shown  in  Fig.  12. 

The  mean  persistency  values  given  in  Fig.  12  show  a  quite  regular 
agreement  with  the  linear  curve  with  a  few  exceptions  at  the  extremes. 


374 


BULLETIN  No.  288 


[April, 


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PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


375 


Persistency  (  k  xio3  ) 
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Initial  Rate  of  Yield  -Lbs.  F.C.M.  per  Pay  (A) 

FIG.  12. — SHOWING  REGRESSION  OF  PERSISTENCY  VALUES  ON  INITIAL 
RATE  OF  YIELD,  GUERNSEY  RECORDS 

Equation  of  the  curve:    y  =  1.782x  —  25.59. 


The  frequencies  here  are  small,  which  may  account  for  the  variation  in 
the  mean  values.  It  is  possible  that  the  entrance  requirements  of  the 
advanced  registry  may  have  some  effect  at  the  lowest  initial  rate  values. 
Thus  a  cow  that  starts  with  a  rate  of  18  pounds  F.C.M.  per  day  would 
need  to  have  a  persistency  of  about  A;  =  0  in  order  to  satisfy  the  mini- 


TABLE  6. — STATISTICAL  CONSTANTS  FOR  INITIAL  RATE  OF  YIELD  AND 
PERSISTENCY:  GUERNSEY  RECORDS 


Constants 

k  X  103 
Persistency 

A 
Initial  rate 
of  yield 
(pounds  F.C.M. 
per  day) 

Mean  

44.25  +.55 

39.19  +.17 

Standard  deviation  

32.19  +  .39 

9.67  +.12 

Coefficient  of  variability  

72.75  +  .89 

24.67  +  .30 

Coefficient  of  correlation  .  . 

.535  - 

h.012 

mum  requirement  of  250.5  pounds  of  fat  at  two  years  of  age  or  under. 
Judging  by  the  data  of  Fig.  12,  no  selective  action  is  apparent  except 
possibly  in  the  case  of  the  two  cows  of  the  18-pound  class. 

The  correction  factors  for  the  persistency  values  are  derived  from 
the  above  equation  k  X  103  =  1.782A  —  25.59,  to  correct  to  the  mean 


376 


BULLETIN  No.  288 


[April, 


value  of  A,  39.19.  Correction  to  this  value  of  A  permits  a  minimum 
correction  to  a  maximum  number  of  the  records.  The  correction  formula 
is  kc  =  k0  +  69.84  —  1.782  A,  where  k0  is  the  observed  k  X  103,  and  kc 
is  the  corresponding  corrected  value. 

The  age-correction  factors  to  be  applied  to  the  yield  for  the  year 
are  derived  from  the  equation  given  in  Fig.  1 1  by  the  direct-ratio  method 
used  by  Gowen8  (chap.  4).  The  age  of  maximum  yield  is  used  as  the  base, 
or  standard  age. 

The  age  correction  factors  for  the  initial  rate  of  yield  are  derived 
from  the  equation  of  Fig.  10  on  the  same  basis  as  above. 

TABLE  7. — MONTH  OF  CALVING  AND  MEAN  VALUES  FOR  PERSISTENCY,  INITIAL  RATE 

OF  YIELD,  YIELD  FOR  YEAR,  AND  FAT  PERCENTAGE: 

GUERNSEY  RECORDS 


A 

fc\S      -t  f\9 

Initial  rate 

Yield  for 

Month 
of 
calving 

Number 
of 
records 

X  ID" 
Persistency 
corrected 
lor  A. 

of  yield 
corrected  for 
age  (pounds 

year  cor- 
rected for 
age  (pounds 

Fat 
percentage 
for  year 

F.C.M.  per 

F.C.M.) 

day) 

January  .... 

95 

48.9 

49.2 

12  700 

5.05 

February  .  .  . 

128 

49.9 

49.0 

12  297 

5.15 

March  

166 

50.0 

52.2 

12  855 

5.05 

April  

158 

50.8 

48.4 

12  411 

5.06 

May  

126 

49,8 

45.6 

12  040 

5.12 

June  

96 

37.3 

44.0 

12  646 

5.01 

July  . 

88 

36.7 

41.2 

12  023 

5.02 

August.1.  .  .  . 

123 

37.0 

42.8 

12  476 

5  03 

September.  . 

134 

37.2 

45.2 

12  731 

5.07 

October  

165 

40.4 

47.6 

12  936 

4.98 

November  .  . 

125 

42.1 

48.4 

12  732 

5.09 

December.  .  . 

130 

52.5 

50.4 

12  508 

5.00 

It  will  be  understood  that  the  correction  of  milk  yield  for  fat 
content  (page  359)  is  really  not  a  correction  in  the  same  sense  as  the 
age  correction.  It  is  rather  a  method  of  estimating  and  expressing  the 
energy  value  of  the  milk. 

Influence  of  Season. — The  season  of  the  year  at  which  the  cow 
calves  may  have  a  considerable  influence  on  her  production  under  ordi- 
nary conditions,  largely  because  of  the  feed  supplied  from  pastures. 
Sanders16  has  shown  that  the  season  of  calving  affects  persistency  under 
the  usual  practice  of  management.  This  seasonal  effect  is  much  less 
important  in  cows  on  official  test  because  of  the  better  and  more  uni- 
form care  and  feed  which  they  receive. 

The  data  are  tabulated  in  Table  7  according  to  the  month  in  which 
the  cow  calved,  and  are  also  given  in  Fig.  13.  It  is  apparent  that  the 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


377 


total  yield  for  the  year  is  but  little  affected  by  the  month  of  calving. 
The  lowest  yields  occur  with  calving  in  May  and  July.  The  effect  of 
the  time  of  calving  on  initial  rate  of  yield  is  somewhat  more  pronounced 
and  regular,  tending  toward  a  low  point  for  July  calvers.  The  persistency 
values  also  show  a  pronounced  drop  in  June,  which  continues  for  the  sum- 
mer months.  Apparently  cows  calving  in  the  summer  months  hold  up 
somewhat  better  than  cows  calving  in  the  winter  months  when  the 
initial  rate  of  yield  is  allowed  for.  No  direct  corrections  are  made  for 
the  effect  of  month  of  calving  on  persistency  or  initial  rate  of  yield. 


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kxi 
A 

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, 

. 

Vie 
fat 

d 
% 

Jan.       Men.      May       July        Sep.       Nov.      Jan. 
Month  of  Calving 

FIG.   13. — EFFECT  OF  THE  SEASON  OF  FRESHENING,  GUERNSEY  RECORDS 

The  initial  rate  of  yield  (A)  in  pounds  of  F.C.M.  per  day  is  corrected  for  age. 
Persistency,  (k  X  103)  rate  of  decrease  per  mille  per  month,  is  corrected  for  A.  The 
yield  for  the  year  in  cwt.  F.C.M.  (Yield)  is  corrected  for  age.  The  fat  percentage  for 
the  year  (Fat  %)  is  taken  directly  from  the  records. 


Variability  in  Persistency. — The  distribution  of  the  observed  per- 
sistency values  has  been  given  in  Table  5,  and  the  constants  derived 
from  this  array  have  been  given  in  Table  6.  The  mean  k  value  is  .04425* ; 

"The  same  records  under  consideration  here  have  been  previously8  dealt  with  in 
groups  classified  on  the  basis  of  the  length  of  the  gestation  included  in  the  record 
period.  It  may  be  noted  that  the  k  values  previously  determined  from  the  group 
records  varied  from  .03714  to  .04361,  and  are  thus  lower  than  the  mean  here  reported. 
The  group  records  include  contemporaneous  reentries.  How  far  the  reentry  records 


378 


BULLETIN  No.  288 


[April, 


standard  deviation,  .03219;  and  coefficient  of  variability,  72.75. 

Correction  of  these  values  on  the  basis  presented  above  leads  to 
the  distribution  given  in  Table  8.  The  two  distributions  are  shown 
graphically  on  a  percentage  basis  in  Fig.  14.  The  corrected  persistency 

TABLE  8. — VARIABILITY  IN  CORRECTED  PERSISTENCY  VALUES  AND  ACCOMPANYING 

VARIABILITY  IN  CALCULATED  YIELD:  GUERNSEY  RECORDS 

(The  persistency  values  have  been  corrected  for  initial  rate  of  yield 

to  A  =  39.19) 


Frequency 

k  X  103 

Persistency 
corrected  for 
A  (class  mid- 
points) 

Yield  for  year 
A  =  39.19 
(pounds 
F.C.M.) 

1  

-  35 

17  821 

3  

-  25 

16  723 

13     

-  15 

15  712 

34              

—     5 

14  779 

79  

5 

13  918 

143  

15 

13  123 

196  

25 

12  389 

226  

35 

11  710 

241  

45 

11  080 

181     

55 

10  498 

161  

65 

9  957 

109  

75 

9  455 

56  

85 

8  98& 

43  

95 

8  556 

25  

105 

8  153 

10  

115 

7  777 

8  

125 

7  427 

0  

135 

2  

145 

6  795 

0  

155 

1  

165 

6  242 

2  

175 

5  992 

Mean  

44.81 

11  260 

Standard  deviation.  .  .  . 
Coefficient  of  variability 

27.32 
60.97 

1  636 
14.53 

may  be  responsible  for  the  lower  k  values  of  the  group  records  in  which  they  appear  has 
not  been  determined.  The  primary  purpose  of  this  note,  however,  is  to  point  out 
that  it  is  not  quite  proper  to  assume  that  the  average  of  the  k's  is  the  same  as  the 
k  obtained  from  the  average  of  the  monthly  data.  As  an  extreme  example,  if  we  have 
two  records  with  the  same  initial  rate  but  in  one  record  k  =  0  and  in  the  other  k  =  .2, 
the  mean  of  the  k's  is  .1;  but  if  the  two  records  are  thrown  together,  the  value  of  k 
determined  from  the  group  record  would  be,  not  .1,  but  about  .06.  Furthermore, 
this  group  lactation  curve  would  not  conform  well  to  the  curve  of  the  exponential 
equation,  having  a  considerably  greater  curvature  than  the  equation  provides. 
Now  the  observed  group  lactation  curves  previously  presented  show  precisely  this 
feature  in  a  very  small  degree.  But  it  would  be  straining  the  point  perhaps  to  ex- 
plain the  deviations  of  the  observed  curves  on  the  basis  of  their  being  made  up  of 
records  which  individually  correspond  to  the  equation  and  therefore  collectively 
differ  from  the  equation  in  the  direction  found. 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


379 


values  give  a  somewhat  more  regular  distribution  curve^than  the  raw 
values.  The  correction  has,  of  course,  caused  a  change  of  places  among 
the  individuals.  The  mean  has  been  slightly  increased  and  the  standard 
deviation  considerably  lowered.  The  coefficient  of  variability  is  still 
very  high  as  compared  with  the  same  constant  for  other  characters.  It 
is  doubtful  if  this  constant  has  its  usual  significance  in  the  present  case. 


14 


" 


0)  & 
01 


-40 


\ 


Cc  Tec  ted 


40  60 

Persistency 


120 


160 


200 


FIG.  14. — PERCENTAGE  FREQUENCY  DISTRIBUTIONS  FOR  PER- 
SISTENCY OF  LACTATION,  GUERNSEY  RECORDS 

The  corrected  k's  have  been  corrected  for  initial  rate  of 
yield  as  explained  in  text. 


Where  the  variability  in  persistency  is  based  on  a  constant  value  of 
initial  rate  of  yield,  the  yields  for  the  year  may  be  computed.  The  last 
column  of  Table  8  gives  these  computed  yields  on  the  basis  of  the  mean 
initial  rate.  These  values  give  14.53  as  the  coefficient  of  variability. 
The  persistency  distribution  of  the  two  initial-rate  arrays  of  Table  5 
at  A  =38  and  40,  by  the  same  treatment  give  a  coefficient  of  variability 


380 


BULLETIN  No.  288 


[April, 


in  yield  of  14.00.  It  appears,  therefore,  that  the  natural  variability  in 
persistency  of  Guernsey  cows  under  the  conditions  and  prescriptions 
of  official  test  is  responsible  for  a  standard  deviation  in  yearly  yield 
equal  to  about  14.5  percent  of  the  mean  yield. 

Variability  in  Initial  Rate  of  Yield. — The  distribution  of  the  initial- 
rate-of-yield  values  has  been  given  ;n  Table  5.    The  coefficient  of 


16 


o!4 

i 
g-1* 

p 

uIlO 

<l> 

0)8 

O 

I* 

u 
|4 


5 


^ 


(BlttCKCiAS 


N 


AS 


16  E4  5a  40  4S  56  64  7£  60  S6  96 

Initial  Rate  of  Yield  (A) 

FIG.    15. — PERCENTAGE    FREQUENCY    DISTRIBUTIONS    FOR 

INITIAL  RATE  OF  YIELD,  GUERNSEY  RECORDS 
The  corrected  A's  have  been  corrected  for  age  of  the  cow 
at  calving. 

TABLE  9. — VARIABILITY  IN  INITIAL  RATE  OF  YIELD  CORRECTED  FOR  AGE: 
GUERNSEY  RECORDS 

(Mean  =  47.22;  standard  deviation  =  9.62;  coefficient  of  variability  =  20.37) 


A«  

fb  

24 
4 

28 
19 

32 
74 

36 
156 

40 
233 

44 
257 

48 
251 

52 
193 

56 
141 

60 

99 

64 
59 

68 
19 

72 
12 

76 
10 

80 
4 

84 
2 

88 
0 

92 
1 

•Class  mid-points  of  initial  rate  of  yield  in  pounds  of  F.C.M.  per  day. 
••Frequency. 

variability  (Table  6)  is  24.67.  Corrected  for  age,  the  values  are  given 
in  Table  9.  The  distributions  for  the  raw  and  corrected  values  are  given 
graphically  on  a  percentage  basis  in  Fig.  15. 

The  mean  initial  rate  is  of  course  increased  by  the  age  correction 
since  the  age  of  maximum  initial  rate  has  been  used  as  a  base.  The  mean 
of  the  age-corrected  data  is  47.22;  the  standard  deviation  is  9.62;  the 
coefficient  of  variability,  20.37. 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


381 


This  coefficient  is  quite  directly  comparable  with  the  same  measure 
of  variability  in  yearly  yield,  since  if  persistency  is  constant  the  yield 
for  the  year  is  proportional  to  the  initial  rate  and  the  coefficients  of 
variability  would  therefore  be  the  same.  The  constants  of  the  age-  and 
fat-corrected  observed  milk  yields  are:  mean,  12,553;  standard  deviation, 
2,503;  coefficient  of  variability,  19.94. 

TABLE  10.— AVERAGE  DAILY  YIELDS  BY  MONTHS  FOR  SPECIAL  GROUP  OF  Cows: 

GUERNSEY  RECORDS 
(Each  record  shows  an  increasing  rate  of  yield  with  advance  in  lactation.) 


Months 
after 
calving 

Number 
of 
records 

Milk  per 
day 
(pounds) 

Fat  per 
day 
(pounds) 

Fat 
percentage 

F.C.M. 

per  day 
(pounds) 

1  

72 

30.0 

1.33 

4.44 

32.0 

2  

83 

28.6 

1.31 

4.59 

31.2 

3  

83 

27.5 

1.31 

4.77 

30.6 

4  

83 

26.9 

1.32 

4.92 

30.6 

5  

83 

26.7 

1.36 

5.09 

31.1 

6  

83 

26.7 

1.38 

5.17 

31.4 

7  

83 

27.1 

1.42 

5.22 

32.1 

8  

83 

27.1 

1.43 

5.27 

32.3 

9  

83 

27.6 

1.46 

5.29 

32.9 

10  

83 

27.5 

1.46 

5.29 

32.8 

11  

83 

26.8 

1.44 

5.37 

32.3 

£30 


29 


Months  after  Calving  (t) 


FIG.  16. — SHOWING  RATE  OF  YIELD  FOR  THE  SPECIAL 
GROUP  OF  Cows,  GUERNSEY  RECORDS 

Each  of  the  83  records  shows  an  increasing  rate  of 
yield  with  advance  in  lactation.  The  equation  of  the 
curve  fitted  by  the  method  of  least  squares  to  the  data 
of  Table  10  is  yd  =  30.725e  o06439'. 


Increasing  Rate  of  Yield  with  Advance  in  Lactation, — Advance  in 
lactation  usually  implies  a  decrease  in  the  rate  of  energy  yield.  A  number 
of  the  records  studied  show  an  increase  in  the  rate  of  energy  yield  with  ad- 
vance in  lactation.  In  Table  5  there  are  83  such  records,  constituting 
5.41  percent  of  the  total.  These  records  are  of  special  interest  because 
of  their  unusual,  if  not  abnormal,  nature.  The  results  from  them, 


382  BULLETIN  No.  288  [April, 

treated  as  a  group,  are  given  in  Table  10  and  Fig.  16.  It  is  evident  from 
Fig.  16  that  these  records  give  an  average  lactation  curve  which  is  not  in 
very  good  conformity  with  the  normal  lactation  curve.  The  first  three 
observations  show  a  normally  decreasing  rate  of  yield,  after  which  there 
is  a  distinct  trend  toward  an  increasing  rate,  and  finally,  in  the  eleventh 
observation,  there  is  evidence  that  a  decreasing  rate  again  has  set  in. 
It  will  be  observed  further  that  the  initial  rate  of  yield  is  much  lower  than 
the  mean  of  all  the  records.  This  relation  implies  the  presence  of  a  dis- 
proportionate number  of  young  cows. 

It  is  evident  that  the  equation  used  does  not  afford  a  complete  de- 
scription of  the  observed  lactation  curve  for  the  records  of  Fig.  10  as  a 
group.  On  the  other  hand,  any  simple  expression  of  the  observed  lacta- 
tion curve  must  recognize  a  general  tendency  for  the  rate  of  yield  to 
increase  with  advance  in  lactation,  and  this  requirement  the  equation 
answers  satisfactorily  for  the  limited  time  range. 

Rate  of  Yield  and  Yield  for  the  Year. — It  is  intended  under  this 
head  to  consider  the  relation  between  the  yearly  yield  and  the  rate  of 
yield  at  various  points  in  the  lactation  curve,  for  various  age  classes,  and 
for  all  ages  together.  We  have  already  seen  (Table  4)  that  there  is  a 
marked  correlation  (r  =  .691)  between  initial  rate  and  yearly  yield,  all 
records  taken  together.  The  question  is,  how  does  this  correlation  be- 
tween rate  and  yearly  yield  change  as  we  consider  the  rate  of  yield  at 
later  times  in  the  lactation;  first,  with  age  relatively  constant;  and 
second,  without  any  age  selection.  Evidently  a  series  of  correlations  may 
be  used  to  indicate  the  common  point  in  the  lactation  curves  at  which 
the  rate  of  yield  is  most  closely  related  to  the  year's  yield.  Such  a  point 
of  closest  relationship  should  be,  theoretically,  the  ideal  time  to  de- 
termine, by  a  short-time  test,  the  probable  year's  production.  Any 
differences  due  to,  or  associated  with  the  age  of  the  cow  should  be 
brought  to  light  by  such  a  series  of  correlations. 

The  rates  of  yield,  determined  by  the  constants  of  the  equations, 
have  been  computed  for  each  lactation  curve  at  ten  additional  points; 
namely,  1,  2,  3,  4,  5,  6,  7,  8,  9,  and  10  months  after  calving.  The  rate  at 
ten  months  after  calving  was  first  computed  algebraically  by  the  use  of 
a  table  of  exponentials.  The  rates  at  the  intermediate  points  were  com- 
puted graphically  by  the  use  of  an  aritho-log  chart  improvised  for  the 
purpose.  The  0  and  10  logarithmic  ordinates  consisted  of  the  scales 
of  a  20-inch  slide  rule.  Connecting  these  two  logarithmic  ordinates,  ab- 
scissas were  drawn  for  the  2-pound  class  intervals  used.  Nine  inter- 
mediate ordinates  were  drawn  to  give  equal  spacing.  A  taut  thread  was 
properly  adjusted  to  the  end  ordinates  according  to  the  two  correspond- 
ing previously-determined  values  for  a  particular  record.  The  points 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


383 


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384 


BULLETIN  No.  288 


[April, 


where  the  thread  cut  the  intermediate  ordinates  served  to  show  the 
corresponding  intermediate  class  values  for  the  lactation  curve. 

The  values  thus  obtained  were  correlated  with  the  observed  yield 
for  the  year,  both  for  all  the  records  together  and  for  various  age  classes. 
The  correlation  coefficients  for  the  various  age  classes  are  given  in 


i.oo 


.80 


<D 

fc 

o 


.60 


40 


'11 


£460 

Months  after  Calving 


10 


FIG.  17. — CORRELATION  BETWEEN  THE  THEORETICAL  RATE  OF  YIELD  AND  THE 
OBSERVED  YIELD  FOR  THE  YEAR,  AGE-CLASSIFIED,  GUERNSEY  RECORDS 

This  is  a  graphic  representation  of  the  data  of  Table  11.  The  figures  at  left  and 
right  of  the  curves  indicate  the  age  classes.  Note  the  rapid  rise  of  the  curves  to  6 
months  and  their  convergence  between  5  and  6  months  after  calving.  Theoretically 
a  short-time  test  will  afford  the  best  index  of  the  year's  production  if  conducted 
during  the  fifth  month  of  lactation.  At  this  point  in  the  lactation  curve  there  is  no 
marked  or  significant  difference  between  the  age  groups.  At  earlier  points  in  the 
lactation  curve  the  rate  of  yield  for  two-year-old  cows  seems  to  be  distinctly  more 
closely  related  to  the  year's  production  than  is  the  case  for  any  other  age. 


Table  11.  It  will  be  observed  from  the  data  of  Table  11  that  there  is 
considerable  variation  in  the  coefficients  at  the  initial  point  of  the  lacta- 
tion curve.  The  highest  correlation,  .728,  is  found  for  two-year-old 
cows;  and  the  lowest,  .485,  for  seven-year-old  cows.  These  relative  posi- 
tions are  maintained  also  at  1,  2,  3,  and  4  months  after  calving.  It 
would  seem,  therefore,  that  any  short-time  test  conducted  shortly  after 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


385 


calving  should  afford  a  better  index  of  the  year's  production  in  the  case 
of  two-year-old,  or  young  cows,  than  in  the  case  of  seven-year-old,  or 
old  cows. 

A  more  remarkable  fact  brought  out  by  Table  11  is  the  very  mater- 
ial increase  in  the  coefficients  at  later  points  on  the  lactation  curve  up  to 


l.oo 


.96 


.92, 


.68 


o 
1 


.84 


0.60 


.76 


.It 


.68 


.(A 


rieorerToJl 


\ 


\ 


10 


2.  A  t>  O 

Months  after  Calving 

FIG.  18. — SHOWING  THE  CORRELATION  BETWEEN  RATE  OF  YIELD  AT  VARIOUS  TIMES 
AFTER  CALVING  AND  YIELD  FOR  THE  YEAR,  ALL  AGES,  GUERNSEY  RECORDS 
The  theoretical  curve  is  based  on  the  fitted  lactation  curves.     The  observed 
curve  is  based  on  the  actual  lactation  curves.    Data  from  Table  17.     The  smooth 
curves  given  in  the  graph  are  merely  free-hand  sketches. 

six  months  after  calving.  At  this  point  on  the  lactation  curve  the  two- 
year-old  group  and  the  seven-year-old  group  have  reached  practically  the 
same  high  value,  .980.  All  of  the  other  age  groups  show  the  highest 
correlation  at  six  months,  except  the  six-year-old  group,  where  the  high- 
est coefficient  is  found  at  five  months.  The  coefficients  decrease  in  value 
after  six  months.  These  relations  for  the  several  age  groups  are  shown 
graphically  in  Fig.  17. 


386 


BULLETIN  No.  288 


[April, 


Similar  correlations  have  been  computed  for  all  ages  lumped  to- 
gether. They  fall  naturally  in  between  the  extremes  given  in  Table  11 
and  Fig.  17.  The  correlation  surface  for  the  theoretical  initial  rate  of 
yield  and  yield  for  the  year  is  given  in  Table  12.  Similar  surfaces  at 
three  months,  six  months,  and  ten  months  after  calving  are  given  in 
Tables  13,  14,  and  15  respectively. 

On  account  of  the  practical  importance  of  the  changes  in  the  close- 
ness of  the  relation  between  the  rate  of  yield  and  yearly  yield  with 


TABLE  12. — CORRELATION  "BETWEEN  THEORETICAL  INITIAL  RATE  OF  YIELD  AND 
TOTAL  YIELD  FOR  THE  YEAR:    GUERNSEY  RECORDS 

Yield  for  year,  hundredweight  F.C.M.:  class  mid-points 


55 

65 

75 

85 

95 

105 

115 

125 

135 

145 

155 

165 

175 

185 

195 

205 

215 

Total 

18 

2 

2 

20 

2 

7 

2 

2 

13 

22 

1 

4 

7 

5 

1 

1 

1 

20 

24 

1 

15 

17 

4 

37 

26 

1 

4 

17 

20 

6 

6 

54 

28 

4 

14 

31 

20 

6 

2 

77 

30 

1 

16 

25 

•}8 

Ifi 

7 

2 

105 

32 

5 

16 

20 

38 

23 

10 

7 

119 

34 

3 

32 

24 

36 

20 

8 

5 

1 

129 

36 

6 

13 

25 

34 

99 

22 

q 

1 

139 

38 

2 

11 

21 

28 

20 

28 

14 

3 

127 

40 

1 

6 

19 

21 

31 

28 

14 

2 

? 

124 

42 
44 

2 

6 

2 

11 
12 

26 
10 

27 
19 

23 
16 

21 
18 

6 
5 

4 
6 

1 

127 

88 

46 

2 

2 

8 

13 

16 

18 

6 

3 

1 

69 

48 
50 
52 
54 
56 
58 
60 
62 
64 

1 
'2 

2 
4 
2 
1 
2 
1 
1 

'i 

13 
6 
11 
2 

'i 

1 

11 
11 
4 
4 
5 
1 
1 
1 
1 

12 
7 
7 
5 
4 
5 
3 
1 

8 
6 
10 
5 
5 
7 
1 
1 
2 

12 
9 
5 
4 
3 
3 
4 
1 
1 

2 
5 
8 
2 
5 

'4 

2 

2 
2 
4 

'4 

2 

2 
3 
1 
1 
2 

'3 
2 

'i 

2 
1 

'2 
1 

65 
54 
54 
24 
26 
24 
24 
10 
5 

66 
68 

2 
? 

'i 

1 

1 

2 

6 
3 

70 

1 

1 

2 

72 

1 

1 

74 

1 

1 

1 

3 

76 

i 

1 

78 

1 

i 

2 

Total 

5 

25 

105 

196 

236 

254 

220 

195 

145 

68 

43 

17 

14 

4 

6 

0 

1 

1534 

advance  in  lactation,  a  similar  comparison  has  been  made  between  the 
observed  monthly  yields  and  yield  for  the  year.  One  correlation  surface 
at  six  months  after  calving  is  given  in  Table  16;  that  is,  the  surface  is 
based  on  the  month  the  mid-point  of  which  is  closest  to  six-months 
after  calving.  The  constants  derived  from  Tables  12  to  16,  and  from 
similar  tables  for  other  points  in  the  lactation  curves,  are  given  hi 
Table  17. 

Table  17  shows  also  the  highest  correlation  between  rate  of  yield 
and  yearly  yield  at  six  months  after  calving.  For  the  theoretical  or 
smoothed  lactation  curve,  the  correlation  at  six  months  after  calving 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


387 


is  .980.  The  corresponding  correlation  for  the  actual  yield  of  the  sixth 
month  and  the  yearly  yield  is  .935.  Both  Comparisons  show  the  maxi- 
mum correlation  at  six  months  after  calving.  The  actual  yield  of  the  sixth 


TABLE  13. — CORRELATION  BETWEEN  THEORETICAL  RATE  OF  YIELD  THREE  MONTHS 
AFTER  CALVING  AND  TOTAL  YIELD  FOR  THE  YEAR:  GUERNSEY  RECORDS 

Yield  for  year,  hundredweight  F.C.M.:  class  mid-points 


55 

65 

75 

85 

95 

105 

115 

125 

135 

145 

155 

165 

175 

185 

195 

205 

215 

Total 

18 

4 

? 

? 

g 

20 

1 

q 

3 

1 

14 

22 

q 

?5 

7 

41 

24 

5 

41 

38 

5 

1 

90 

26 

23 

e>?, 

?fi 

3 

1 

115 

28 

6 

55 

70 

16 

1 

148 

30 

4 

22 

fiO 

53 

10 

149 

32 

1 

8 

4?: 

7? 

30 

9 

162 

34 

?, 

?1 

51 

53 

?0 

1 

148 

36 

8 

?6 

63 

54 

13 

1 

165 

38 

1 

3 

?1 

3fi 

53 

34 

5 

153 

40 

8 

15 

3? 

45 

7 

1 

108 

42 

1 

2 

7 

13 

20 

13 

8 

64 

44 

2 

12 

15 

19 

6 

1 

55 

46 

2 

2 

13 

12 

11 

2 

1 

43 

48 

1 

2 

7 

11 

R 

3 

30 

50 

2 

2 

4 

2 

4 

14 

52 

1 

2 

1 

9 

54 

5 

1 

6 

56 

1 

1 

3 

2 

7 

58 

2 

2 

60 

1 

1 

62 

0 

64 

1 

1 

2 

Total 

5 

25 

105 

196 

236 

254 

220 

195 

145 

68 

43 

17 

14 

4 

6 

0 

1 

1534 

TABLE  14. — CORRELATION  BETWEEN  THEORETICAL  RATE  OF  YIELD  Six  MONTHS 
AFTER  CALVING  AND  TOTAL  YIELD  FOR  THE  YEAR:  GUERNSEY  RECORDS 

Yield  for  year,  hundredweight  F.C.M.:  class  mid-points 


55 

65 

75 

85 

95 

105 

115 

125 

135 

145 

155 

165 

175 

185 

195 

205 

215 

Total 

16 

5 

3 

1 

1 

10 

18 

?1 

q 

30 

20 

1 

54 

3 

58 

22 

39 

86 

4 

129 

24 

2 

103 

49 

1 

155 

26 

4 

155 

27 

1 

187 

28 

?6 

149 

q 

184 

30 

1 

75 

96 

172 

32 

1 

105 

31 

1 

138 

34 

1 

9 

134 

g 

152 

36 

?q 

77 

1 

107 

38 

1 

55 

30 

86 

40 

5 

32 

9 

46 

42 

4 

23 

1 

28 

44 

11 

9 

20 

46 

6 

fi 

12 

48 

1 

7 

8 

50 

1 

? 

3 

52 

2 

1 

3 

54 

? 

2 

56 

3 

3 

58 

0 

60 

i 

1 

Total 

5 

25 

105 

196 

236 

254 

220 

195 

145 

68 

43 

17 

14 

4 

6 

0 

1 

1534 

388 


BULLETIN  No.  288 


[April, 


TABLE  15. — CORRELATION  BETWEEN  THEORETICAL  RATE  OF  YIELD  TEN  MONTHS 
AFTER  CALVING  AND  TOTAL»YIELD  FOR  THE  YEAR:    GUERNSEY  RECORDS 

Yield  for  year,  hundredweight  F.C.M.:  class  mid-pointa 


55 

65 

75 

85 

95 

105 

115 

125 

135 

145 

155 

165 

175 

185 

195 

205 

215 

Total 

g 

1 

1 

•  g 

2 

2 

10 

1 

2 

1 

1 

5 

j 

3 

5 

4 

3 

1 

17 

14 

3 

9 

18 

g 

3 

3 

1 

45 

16 

3 

28 

26 

13 

7 

1 

78 

18 

7 

34 

52 

19 

12 

4 

128 

20 

1 

16 

46 

49 

32 

7 

2 

153 

22 

2 

47 

75 

49 

25 

6 

1 

205 

24 

8 

46 

53 

33 

13 

1 

1 

155 

26 

4 

21 

61 

57 

22 

11 

176 

28 

4 

28 

44 

29 

10 

3 

1 

119 

30 

1 

7 

30 

42 

28 

11 

119 

32 

1 

12 

47 

32 

10 

5 

107 

34 

6 

?<) 

22 

16 

5 

78 

36 

3 

?5 

14 

6 

f, 

50 

38 

?! 

11 

7 

10 

3 

1 

34 

40 

3 

4 

10 

2 

1 

20 

42 

1 

? 

5 

6 

4 

18 

44 

2 

2 

5 

1 

10 

46 

1 

3 

1 

? 

7 

48 

f 

2 

50 

3 

3 

52 

0 

54 

1 

1 

56 

1 

1 

Total 

5 

25 

105 

196 

236 

254 

220 

195 

145 

68 

43 

17 

14 

4 

6 

0 

1 

1534 

TABLE  16. — CORRELATION  BETWEEN  AVERAGE  DAILY  YIELD  FOR  SIXTH  CALENDAR 
MONTH  OF  RECORD  AND  TOTAL  YIELD  FOR  THE  YEAR:  GUERNSEY  RECORDS 

Yield  for  year,  hundredweight  F.C.M.:  class  mid-points 


55 

65 

75 

85 

95 

105 

115 

125 

135 

145 

155 

165 

175 

185 

195 

205 

215 

Total 

14 

1 

1 

16 

3 

3 

4 

10 

18 

1 

13 

13 

4 

31 

20 

8 

38 

?3 

? 

71 

22 

1 

1 

35 

77 

10 

1 

125 

24 

^?, 

fi« 

55 

6 

139 

26 

?, 

18 

103 

57 

5 

3 

188 

28 

6 

50 

74 

?6 

?, 

1 

159 

30 

?, 

11 

73 

R7 

10 

3 

166 

32 

ft 

35 

61 

47 

4 

1 

153 

34 

8 

46 

fi? 

?? 

? 

140 

36 

10 

47 

41 

5 

1 

104 

38 

1 

4 

18 

38 

?3 

3 

87 

40 

4 

21 

12 

5 

42 

42 

1 

11 

13 

14 

4 

43 

44 

1 

4 

6 

12 

g 

29 

46 

4 

5 

4 

'g 

'i 

22 

48 

2 

1 

3 

6 

50 

2 

1 

2 

3 

g 

52 

2 

1 

3 

54 

1 

1 

2 

56 

2 

2 

58 

2 

2 

60 

0 

62 

0 

64 

1 

1 

Total 

5 

25 

105 

196 

236 

254 

220 

195 

145 

68 

43 

17 

14 

4 

6 

0 

1 

1534 

1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


389 


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Observe 

390  BULLETIN  No.  288  [April, 

month  is  less  closely  related  to  the  year's  production,  than  is  the  cal- 
culated yield  for  that  month.  This  is  not  surprising  because  of  the 
considerable  irregularities  of  the  realized  lactation  curve.  The  con- 
dition is  accentuated,  furthermore,  by  the  fact  that  the  fat  test  for  the 
month  represents  only  two  days,  or  one  day  in  a  minority  of  the  records, 
thus  increasing  the  variability  of  the  F.C.M.  observations.  It  is  probable, 
in  other  words,  that  similar  comparisons  based  on  the  raw  milk  yields 
would  show  higher  correlations  than  those  found  for  the  F.C.M.  values. 
One  such  correlation  for  raw  milk  yields  has  been  computed  for  the 
6th  month  and  gives  r  =  .938  +  .002. 

The  correlation  coefficients  for  the  theoretical  and  observed  lac- 
tation curves  are  given  graphically  in  Fig.  18.  It  is  evident  that  the  cor- 
relations change  in  a  quite  regular  manner  with  advance  in  lactation, 
reaching  a  maximum  during  the  fifth  month  of  lactation.  There  is  a 
reason  for  this  in  terms  of  the  theoretical  lactation  curves. 

We  have  seen  above  (Fig.  4),  with  reference  to  the  theoretical 
lactation  curves,  that  the  yields  for  the  year  are  proportional  to  the 
ordinates  of  the  curves  at  any  specified  time,  provided  k  is  constant. 
Theoretically,  therefore,  the  only  reason  why  there  is  not  perfect  cor- 
relation between  the  computed  rate  of  yield  and  the  observed  yearly 
yield  is  because  of  variability  in  persistency.  We  must  add  to  this 
reason  the  failure  to  adjust  perfectly  the  theoretical  lactation  curves  to 
the  observed  yields,  and  also  some  variability  in  the  time  after  calving 
at  which  the  record  starts.  The  influence  of  variability  in  persistency, 
in  affecting  the  correlation  between  the  rate  of  yield  at  various  times  after 
calving  and  the  total  yield  for  twelve  months,  may  be  approached  as 
follows: 

dij 

Consider  the  lactation  curve  as  represented  by  equation  (1),  -f  = 

at 

ae~kt,  the  area  under  which,  from  t  =  0  to  t  =  12,  representing  the 
total  yield  for  twelve  months,  is  j-  (1  —  e~12*).  It  is  clear  from  Fig.  5 

K 

that  if  we  have  given  a  point  on  the  lactation  curve  at  the  origin,  t  =  0, 
the  area  under  the  curve  will  vary  inversely  with  the  value  of  k.  On  the 
other  hand,  if  we  have  given  a  point  on  the  lactation  curve  at  the 
opposite  end,  t  =  12,  the  area  will  vary  directly  with  k.  At  some  point 
in  the  curve  between  t  =  0  and  t  =  12,  the  change  in  area  with  change 
in  k  passes  from  inverse  to  direct.  Let  this  point  be  designated  t' ;  that 
is,  t'  is  an  assumed  point  in  the  lactation  curve  where  a  change  in  A; 
produces  no  change  in  area.  Let  Y  =  the  area,  or  twelve  months '  yield,  = 

,  (1  —  e~12A);  and  let  a'  =  the  rate  of  yield  at  the  time  t',  =  ae~kt'. 


19S7J 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


391 


Then  a  =  aV"  and  Y  = 


a  e' 


(1  —  e~12Ar).      We  may  determine  the 

A/ 

value  of  t'  by  differentiating  Y  with  respect  to  k  and  setting  -—    =  0, 

ak 


whence 


dY 

~dk 


>0kt' 


a  e 


+  (1  - 


ka't'ek"  -  a'ek" 


1          12e~12fc 
=  0,  and  t'  =  -  -  —    — — 

k        I  -*-  e~12fc 


fc        1  -  e-12*_ 

Solving  for  £'  at  the  various  class  values  of  k  found  in  the  data,  we 
have  the  results  given  in  Table  18.  It  is  evident  that  t'  is  not  a  fixed 
value  but  varies  with  the  value  of  k.  Applying  the  frequencies  of  Table 
8  to  the  t'  values  of  Table  18  leads  to  the  constants:  mean  =  5.476  ± 
.006;  standard  deviation  =  .373  ±  .005;  coefficient  of  variability 
=  6.81  +  .08. 

For  the  k's  actually  found,  therefore,  there  is  a  point  in  the  lacta- 
tion curves,  t  =  5.476,  at  which  variation  in  k  has  little  or  no  effect  on 

TABLE  18. — SHOWING  THE  POINTS,  t',  IN  THE  LACTATION  CURVES  AT  WHICH  THE 

YIELD  FOR  THE  YEAR  Is  NOT  AFFECTED  BY  CHANGES  IN  PERSISTENCY,  k 
(The  A;  values  are  the  various  class  values  represented  in  the  Guernsey  records) 


k  X  103.. 

-35 

-25 

-15 

-5 

5 

15 

25 

35 

45 

t'  

6.42 

6.30 

6.18 

6.06 

5.94 

5.82 

5.70 

5.58 

5.46 

A;  X  103.. 

55 

65 

75 

85 

95 

105 

115 

125 

135 

t'  

5.35 

5.23 

5.11 

5.00 

4.88 

4.77 

4.66 

4.55 

4.45 

k  X  103.  . 

145 

155 

165 

175 

185 

195 

205 

215 

t'  

4.34 

4.24 

4.14 

4.04 

3.94 

3.85 

3.76 

3.67 

the  area.  The  rate  of  yield  at  this  point  should  be  directly  proportional 
to  the  yield  for  the  year  regardless  of  the  persistency  value.  By  inference 
we  accordingly  may  expect  to  find  the  highest  degree  of  correlation  be- 
tween rate  of  yield  and  the  yield  for  the  year  at  or  close  to  this  point. 
This  expectation  is  realized  in  the  actual  observations.  Since  the  observed 
records  start  a  few  days  after  calving,  there  would  be  a  tendency  for 
the  highest  observed  correlation  to  appear  slightly  later  than  the  com- 
puted time. 

The  constants  of  Table  17  afford  the  basis  for  estimating  the  yearly 
yield  from  the  rate  of  yield  per  day.  The  equation  from  the  smoothed 
lactation  curves  at  six  months  after  calving  is  y  =  (354.40z  +  421.8)  +  318; 
and  from  the  actual  yield  of  the  sixth  month  y  =  (317.02x  -f-  1431.3) 
±  570.  The  probable  errors  are  for  a  single  estimate. 

Another  point  of  interest  in  connection  with  Table  17  is  in  the 
mean  values  of  the  observed  and  calculated  rates  of  yield.  A  fairly  good 


392  BULLETIN  No.  288  [April, 

agreement  is  evident,  but  it  appears  that  the  k  values  have  been  slightly 
overestimated.  That  is,  the  calculated  lactation  curves  average  a  slightly 
greater  rate  of  decrease  than  is  shown  by  the  average  actual  data.  Com- 
pare Fig.  6  in  this  connection. 

Influence  of  Heredity  and  Environment. — It  is  well  recognized  by 
common  observation  that  the  yearly  production  of  cows  is  influenced 
both  by  their  ancestry  and  by  the  feed  and  care  which  they  receive. 
Quantitative  measures  of  the  influence  of  heredity  and  environment  on 
milk  yield  and  fat  percentage  have  been  derived  by  statistical  methods, 
and  published  (cf.  particularly  Gowen,8  chap.  19).  Similar  methods  are 
adopted  here  to  study  the  relative  effect  of  heredity  and  environment  on 
persistency  of  lactation.  Initial  rate  of  yield,  yield  for  the  year,  and 
fat  percentage  are  also  considered  in  the  same  connection  as  affording 
data  of  value  and  as  a  check  on  the  results  shown  for  persistency. 

The  records  were  sorted  into  herd  groups  by  the  owner's  name. 
Herds  were  selected  which  contained  at  least  two  unrelated  cows,a  and 
at  least  two  cows  by  the  same  sire  and  from  different  dams. 

This  condition  was  met  by  72  herds  containing  252  half-sisters  by 
97  sires,  and  273  unrelated  cows.  The  records  of  the  cows  with  respect  to 
persistency,  initial  rate  of  yield,  yield  for  the  year,  and  fat  percentage 
have  been  correlated  by  the  method  of  Harris.12  The  method  is  equiva- 
lent to  using  all  possible  combinations  of  the  variables  correlated,  and 
gives  a  total  number  for  half-sisters  of  532;  and  for  unrelated  cows, 
1,224.  The  derived  constants  are  given  in  Table  19. 

Corresponding  correlations  have  also  been  derived  for  half-sisters 
from  a  common  dam  and  by  different  sires;  for  full  sisters;  and  for  cows 
related  as  dam  and  daughter.  The  three  combinations  just  mentioned 
were  limited  tp  the  same  herd,  but  on  account  of  the  limited  number  of 
such  combinations  all  herds  were  included  in  which  there  were  two  or 
more  cows  of  any  of  the  relationships  specified.  The  half-sister  (common 
dam)  combinations  total  48;  full-sister,  38;  and  dam-daughter,  54.  The 
derived  constants  are  given  also  in  Table  19. 

Since  we  are  dealing  now  with  smaller  groups,  there  is  more  chance 
that  any  group  may  not  be  entirely  representative.  The  values  of  the 
means  given  in  Table  19  afford  some  indication  as  to  this.  It  will  be 
seen  that  there  is  fairly  good  agreement  between  the  means  for  the  several 
groups  except  in  the  case  of  the  full-sister  group.  In  this  group  both  the 
initial  rate  of  yield  (A)  and  decrease  in  rate  of  yield  (ft)  are  high.  Also 
the  fat  percentage  of  the  full-sister  group  is  distinctly  above  the  average 
of  all  the  records.  There  is  consequently  some  question  as  to  the  repre- 
sentativeness of  the  full-sister  group. 

'By  "unrelated"  is  meant  that  the  cows  were  not  related  as  half-sisters,  full 
sisters  or  dams  and  daughters. 


1927]  PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows  393 

The  coefficients  of  correlation  given  in  the  last  section  of  Table 
19  afford  an  indication  of  the  relative  influence  of  heredity  and  en- 
vironment, within  the  range  covered,  on  the  performance  of  the  cow. 
The  coefficients  for  the  unrelated  herd  mates  show  the  degree  of  resem- 
blance in  performance  due  to  similarity  of  environmental  factors  within 
a  single  herd  and  to  dissimilarity  of  such  factors  as  between  the  several 
herds  represented.  The  resemblance  may  be  due  in  part,  also,  to  blood 
relationship  more  remote  than  that  of  half-sister. 

Considering  the  data  for  fat  percentage,  which  is  well  recognized 
as  an  individual  and  breed  characteristic,  it  will  be  noted  at  once  that 
daughters  of  a  given  sire  within  a  particular  herd  tend  to  resemble  each 
other  more  closely  (r  =  .305)  than  do  unrelated  cows  within  a  particular 
herd  (r  =  .138).  It  will  be  recalled  from  the  grouping  used  that  each  of 
the  72  herds  is  represented  by  at  least  two  unrelated  cows  and  by  two 
half-sisters.  It  seems  proper,  therefore,  to  attribute  the  greater  re- 
semblance of  the  half-sisters  to  the  additional  effect  of  the  common  par- 
entage on  the  sire's  side. 

A  more  definite  measure  of  the  influence  of  the  sire  may  be  obtained 
by  the  path  coefficient  method  of  analysis  of  Wright.21  It  may  be  assumed 
that  the  correlation  between  a  half-sister  and  an  unrelated  herd  mate 
would  be  the  same  as  between  unrelated  herd  mates.  If  we  let  1  and  2 
stand  for  related  cows  and  3  and  4  for  unrelated  cows,  then  with  re- 
spect to  fat  percentage  from  Table  18,  n2  =  .305  and  r34  ( =  rJ3  =  r23  = 
TU  =  7*24)  =  .138.  The  coefficient  ru  is  determined  by  the  action  of  two 
forces:  the  common  herd,  or  environment,  and  the  common  parentage, 
or  heredity.  Representing  these  forces  by  E  and  //respectively,  the  re- 
sult is  shown  diagramatically  in  Fig.  19.  The  correlation  between  half- 
sisters  is  ri2  =  .3054  =  h2  -f  e2;  the  correlation  between  unrelated  cows, 
r34  =  .1376  =  e2.  Hence  h2  =  .3054  -  .1376  =  .1678.  The  correlation 
between  half-sisters  with  the  herd  influence  eliminated,  r^.E  becomes, 

1^2  I  ft7<2 

— -  =  —  —  =  .1946.  This  is  the  correlation  to  be  expected  be- 
tween half-sisters  on  the  sire 's  side  within  a  single  large  herd  or  where 
the  herd  conditions  are  the  same.  Additional  similar  coefficients  are 
given  in  Table  20. 

It  may  be  seen  from  Table  19  that  half-sisters  from  a  common  dam 
show  a  similar  degree  of  resemblance  with  respect  to  fat  percentage  as 
that  shown  by  half-sisters  by  a  common  sire.  The  closer  relationship, 
represented  by  full  sisters  or  dam  and  daughter,  shows  a  higher  degree 
of  resemblance. 


394 


BULLETIN  No.  288 


[April, 


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TABLE  19.  —  STATISTICAL  CONST.A 

Intragroupal  relation 

Unrelated  herd  mates8  
Half-sister  herd  mates  (C-S)b  
Half-sister  herd  mates  (C-D)C  
Full-sister  herd  mates  

Dam-daughter  herd  mates  /dams. 
I  daught 

Unrelated  herd  mates  
Half-sister  herd  mates  (C-S)  
Half-sister  herd  mates  (C-D)  
Full-sister  herd  mates  
Dam-daughter  herd  mates.  .  .  .  /dams.  . 
Idaught 

"Not  related  as  dam  and  daughter,  j 

PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


395 


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Unrelated  herd  mates*  
Half-sister  herd  mates  (C-S)b  
Half-sister  herd  mates  (C-D)C  
Full-sister  herd  mates  *.  .  .  . 

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Idai 

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Half-sister  herd  mates  (C-S)  
Half-sister  herd  mates  (C-D)  
Full-sister  herd  mates  
Dam-daughter  herd  mates  

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396 


BULLETIN  No.  288 


[April, 


With  respect  to  persistency  of  lactation  it  appears  from  Table  20 
that  there  is  no  correlation  between  half-sisters  by  a  common  sire;  but 
there  is  a  considerable  correlation  between  half-sisters  from  a  common 
dam.  There  is  also  a  significant  correlation  between  dam  and  daughter. 


FIG.  19.  —  ANALYSIS  OF  THE  CORRELATION  BETWEEN  RELATED  HERD  MATES,  AS 
DETERMINED  BY  HEREDITY  AND  ENVIRONMENT 

Unrelated  herd  mates  are  represented  by  3,  4;  related  herd  mates,  by  1,  2.  It  is 
assumed  that  the  correlation  between  unrelated  herd  mates  (m)  is  the  result  of  the 
common  environment  (E);  the  correlation  between  related  herd  mates  (ri2)  is  the  re- 
sult of  the  common  environment  (E)  plus  the  common  parentage  or  heredity  (H): 


rs4  =  e2;  ri2  = 
relationship. 


e2  +  h2;  rJ2  .E  = 


the  correlation  associated  with  the  blood 


TABLE  20.  —  ESTIMATED  COEFFICIENTS  OF  CORRELATION  WITH  RESPECT  TO  CERTAIN 

CHARACTERS,  BETWEEN  Cows  OF  VARIOUS  BLOOD  RELATIONSHIPS  WITHIN 

AN  INDEFINITELY  LARGE  HERD  :    GUERNSEY  RECORDS 


^) 

Relationship 

Characters 

Persistency 
of  lactation 
corrected  for 
A 

Initial  rate 
of  yield  cor- 
rected for 
age 

Yield  for 
year  cor- 
rected for 
age 

Fat 
percentage 
for  year 

Estimated  coefficients  of  correlation 

Half-sister,  common  sire.  .  . 
Half-sister,  common  dam  .  . 
Full-sister  

-.001  +.029 
.418  +  .080 
-.178  ±.106 
.259  +.086 

.186  +  .028 
.093  +  .097 
.387  ±  .093 
.458  +  .073 

.273  +  .027 
.424  +  .080 
.169  ±  .106 
.364  ±  .080 

.195  +  .028 
.156  +  .095 
.329  ±  .098 
.313  ±  .083 

Dam-daughter  

These  relations  suggest  the  interesting  hypothesis  that  the  persistency 
of  a  cow  is  affected  by  inheritance  thru  the  dam  but  not  thru  the  sire. 
The  correlation  between  full  sisters  does  not  bear  out  such  an  hypothesis 
and  in  view  of  the  large  probable  errors  of  the  coefficients  it  may  be 
questioned  whether  such  an  interpretation  is  justified.  On  the  other 
hand,  it  has  been  noted  above  that  the  full-sister  group  does  not  conform 
closely  to  the  average  of  all  the  records.  Also,  the  correlation  between 
full  sisters  with  respect  to  yield  for  the  year  (r  =  .169)  is  less  than  is  to 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


397 


be  expected  from  other  results  (Gowen,8  p.  145).  It  seems  clear  that 
there  is  no  resemblance  between  half-sisters  on  the  sire's  side  with 
respect  to  persistency  of  lactation.  This  means  either  that  the  sire 
exerts  no  influence  on  his  daughters  or  that  all  exert  the  same  influence. 


RESULTS  FROM  HOLSTEIN  RECORDS 

The  Holstein  data,  in  so  far  as  they  permit,  will  be  presented  in  the 
same  general  fashion  as  has  been  used  for  the  Guernsey  records.  The 
Holstein  data  do  not  include  any  record  of  the  year's  performance  or 
date  of  freshening,  so  that  relations  involving  these  items  cannot  be 
considered.  They  do  include  one  item  not  included  in  the  Guernsey 
data,  namely,  the  performance  of  the  same  cow  in  different  lactations. 
There  is,  of  course,  a  large  amount  of  material  of  this  kind  available  in 
the  published  Guernsey  records,  but  the  present  study  has  been  con- 
fined to  original  entries  in  the  Guernsey  records. 

Age  and  Persistency. — The  mean  persistency  values  for  the  various 
age  classes  are  given  in  Table  21  and  Fig.  20.  The  mean  values,  like 
those  for  the  Guernsey,  are  quite  irregular,  and  a  satisfactory  repre- 
sentation by  a  smooth  curve  is  difficult.  It  is  clear,  however,  that  there 
is  a  distinct  rise  in  the  curve  at  first,  followed  by  a  less  rapid  decline. 

TABLE  21. — VARIOUS  AGE  CLASSES  AND  THE  CORRESPONDING  MEAN  VALUES  FOR 
PERSISTENCY  AND  THEORETICAL  INITIAL  RATE  OF  YIELD:  HOLSTEIN  RECORDS 


Age  in  years 
(class  mid- 
points) 

Number 
of 
records 

k  X  103 
Persistency 

A 
Initial  rate  of 
yield  (pounds 
F.C.M.  per 
week) 

1  75  ...                    

48 

45.8 

339.2 

2.25  

306 

43.2 

377.3 

2  75  ...                   

193 

47.1 

415.8 

3.25  

136 

56.6 

461.8 

3.75  

119 

58.9 

491.1 

4.25  

84 

62.4 

528.1 

4  75      ...               

98 

57.0 

524.9 

5.25  

82 

62.2 

542.9 

5.75  

66 

57.1 

547.9 

6.25  

66 

63.2 

561.2 

6.75  

51 

53.2 

526.3 

7.25  

33 

58.0 

592.7 

7.75  

18 

51.1 

520.0 

8.25  

25 

62.2 

582.4 

8.75  

14 

55.7 

580.0 

9.25  

13 

57.3 

606.2 

9.75  

18 

61.7 

586.7 

10.5  

12 

50.8 

536.7 

11.5  

10 

55.0 

564.0 

12.5  

2 

60.0 

560.0 

13.5  

1 

75.0 

640.0 

398 


BULLETIN  No.  288 


[April, 


As  compared  with  the  Guernsey  curve  (Fig.  9),  the  changes  with  age  are 
somewhat  less  pronounced. 

Age  and  Initial  Rate. — The  mean  values  for  initial  rate  of  yield  per 
week  also  are  given  in  Table  21.    Graphic  presentation  is  given  in  Fig. 


80 


70 


40 


50 


*         a         to 

Age  -Years 


ia 


ILLUSTRATING  THE  CHANGE  IN  PERSISTENCY  WITH 
ADVANCING  AGE,  HOLSTEIN  RECORDS 


FIG.  20. 


Equation  of  the  curve:  y  =  35.6  -  8.36x  +  . 
88.73  log  x,  x  being  the  age  in  years.  According  to  the  equa- 
tion,  y  reaches  a  maximum  at  x  =  6.09  =  6  years,  1  month, 
3  days,  at  which  age  y  =  60.5.  That  is,  the  rate  of  decrease 
in  the  rate  of  yield  per  month  at  that  age  is  60.5  per  mille 
per  month. 


21.  The  changes  in  rate  of  yield  with  age  are  quite  regular  up  to 
years,  after  which  there  is  considerable  irregularity  due  in  part  to  the 
small  numbers  represented.  As  compared  with  the  similar  data  for 
Guernseys  (Fig.  10),  the  changes  with  age  are  much  more  pronounced. 
It  should  be  noted  that  the  vertical  scale  of  Fig.  21  is  about  half  that 
of  Fig.  10. 

Persistency  and  Initial  Rate.  —  The  relation  of  persistency  to  initial 
rate  of  yield  and  to  age  may  be  shown  approximately  by  dealing  with  the 
first  part  of  the  age  curve  in  order  to  secure  a  reasonable  approach  to 
linear  regression.  For  this  purpose  we  may  consider  the  records  of  those 
cows  under  4*^  years  of  age  at  calving.  There  are  886  such  records, 
making  up  63.5  percent  of  the  total,  and  the  constants  are  given  in 
Tables  22  and  23. 

Principal  interest  attaches  to  the  correlation  between  persistency 
and  age  with  initial  rate  constant,  the  partial  coefficient  of  correlation 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


399 


being  .006.  This  agrees  with  the  Guernsey  results  in  indicating  that 
persistency  is  independent  of  age  except  as  the  initial  rate  of  yield  is 
associated  with  age. 

The  correlation  surface  for  persistency  and  initial  rate  of  yield  is 
given  in  Table  24  and  the  constants  derived  from  this  surface  are  pre- 


o 

?550 

a. 

2500 
<ri 

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0) 

^  400 

1 

J2350 
'E 

*>oo 

o 

o 

o 

o   o 

0 

^ 

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o 

0 

«/ 

/ 

o 

X 

x 

7 

0 

/ 

/ 

y 

/ 

7 

" 

J 

I 

a            4           ^            e            10           la           M 
Age  -Years 

FIG.  21. — SHOWING  THE  CHANGE  IN  INITIAL  RATE  OP  YIELD 
WITH  ADVANCING  AGE,  HOLSTEIN  RECORDS 

Equation  of  the  curve:  y  =  213.5  —  23.50a;  —  .546s2  + 
645.78  log  x,  x  being  age  in  years.  This  gives  y  a  maximum 
value  at  x  —  8.542  =  8  years,  6  months,  15  days,  at  which  time 
y  =  574.4,  the  maximum  initial  rate  of  yield  in  pounds  F.C.M. 
per  week. 


sented  in  Table  25.  The  correlation  (r  =  .433)  is  not  quite  as  high  as 
that  found  for  the  Guernsey  records  (r  =  .535).  The  mean  persistency 
value,  k  X  103  =  53.2,  shows  a  considerably  greater  rate  of  decline 
than  was  shown  by  the  Guernsey  records.  In  this  connection  it  is 
necessary  to  consider  the  initial  rate  of  yield — which  for  the  Holstein 
records  is  471  pounds  of  4-percent  milk  per  week,  or  67.3  pounds  per  day 
— as  compared  with  39.2  pounds  of  4-percent  milk  per  day  for  the 
Guernsey  data.  Thus  while  at  first  sight  it  might  seem  that  the  Hol- 
stein cows,  as  compared  with  the  Guernsey,  show  a  greater  rate  of  de- 


400 


BULLETIN  No.  288 


[April, 


TABLE  22. — STATISTICAL  CONSTANTS  OF  PERSISTENCY  AND  INITIAL  RATE  OF  YIELD 
FOR  AGES  UNDER  FOUR  AND  ONE-HALF  YEARS:    HOLSTEIN  RECORDS 


Constant 

k  X  10s 
Persistency 

A 
Initial  rate 
(pounds 
F.C.M.  per 
week) 

50.2    +    .7 

426.1     +  2.3 

Standard  deviation  

28.9    +    .5 

99.3    +  1.6 

Coefficient  of  variability,  

57.58  +  .92 

23.31  +     .37 

TABLE  23. — SIMPLE  AND  PARTIAL  COEFFICIENTS  OF  CORRELATION  BETWEEN  PER- 
SISTENCY, INITIAL  RATE  OF  YIELD,  AND  AGE:  HOLSTEIN  RECORDS  UNDER 
FOUR  AND  ONE-HALF  YEARS 


Variables  correlated 

Variables  constant 

Coefficients 

Persistency  and  initial  rate  

.407  +  .019 

Persistency  and  age  

.232  +    021 

Initial  rate  and  age  

.559  +    016 

Persistency  and  initial  rate  

Age  

.344  +  .020 

Persistency  and  age  

Initial  rate  

.006  +  .023 

Initial  rate  and  age  

Persistency  

.523  +  .017 

TABLE  24. — CORRELATION  SURFACE  FOR  INITIAL  RATE  OF  YIELD  AND  PERSISTENCY: 

HOLSTEIN  RECORDS 

Initial  rate  of  yield,  pounds  F.C.M.  per  week  (A):    class  mid-points 


240 

280 

320 

360 

400 

440 

480 

520 

560 

600 

640 

680 

720 

760 

800 

840 

880 

Total 

-25 

1 

1 

1 

3 

-15 

? 

6 

1 

..  8 

-  5 

7 

3 

3 

? 

1 

1 

1 

18 

5 

7 

10 

7 

R 

11 

? 

1 

4 

48 

15 

1 

5 

17 

1?, 

13 

8 

6 

5 

5 

1 

73 

25 

5 

15 

23 

24 

16 

15 

15 

15 

3 

5 

3 

4 

143 

35 

1 

8 

17 

22 

18 

30 

20 

14 

6 

10 

4 

1 

1 

1 

153 

45 

5 

16 

26 

29 

38 

30 

29 

10 

5 

6 

7 

5 

2 

208 

55 

2 

6 

23 

31 

24 

31 

29 

20 

13 

10 

4 

7 

3 

2 

205 

65 

8 

4 

14 

18 

20 

21 

23 

17 

?,?, 

8 

8 

4 

?, 

1 

1 

161 

75 

1 

3 

6 

10 

17 

25 

22 

18 

16 

16 

7 

2 

2 

1 

146 

85 

2 

5 

4 

8 

19 

15 

15 

11 

13 

8 

?, 

?, 

8 

1 

109 

95 

3 

6 

9 

12 

10 

6 

8 

3 

3 

1 

55 

105 

1 

1 

2 

5 

7 

3 

4 

2 

1 

2 

i 

29 

115 

2 

1 

1 

2 

3 

2 

1 

2 

1 

15 

125 

1 

3 

2 

2 

1 

1 

1 

11 

135 

1 

3 

4 

145 

1 

1 

2 

155 

0 

165 

1 

1 

2 

175 

1 

1 

2 

Total 

17 

64 

101 

147 

165 

183 

181 

182 

109 

95 

62 

33 

31 

13 

8 

3 

1 

1395 

crease  in  yield  with  advance  in  lactation,  the  very  marked  difference  in 
level  of  production  may  require  a  modification  of  such  a  view. 

The  Holstein  lactation  curves,  it  will  be  remembered,  are  based  on 
two  7-day  tests;  the  second  test  occurring  not  less  than  eight  months 
after  calving.  It  would  seem  that  there  might  be  a  very  natural  tendency 
to  conduct  this  second  test  only  on  those  cows  which  were  milking  heav- 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


401 


ily  enough  to  make  a  creditable  record,  that  is,  to  exclude  cows  of  more 
rapid  decrease  in  rate  of  yield.  Such  a  practice  would  tend  to  make  the 

TABLE  25. — STATISTICAL  CONSTANTS  FOR  INITIAL  RATE  OF  YIELD  AND  PERSISTENCY  : 

HOLSTEIN  RECORDS 


Constant 

kX  103 
Persistency 

A 
Initial  rate 
of  yield 
(pounds  F.C.M. 
per  week) 

Mean  

53  2  +    5 

471  +21 

Standard  deviation  

28  1  +    4 

116  +15 

Coefficient  of  variability  

52.85  +    67 

24  63  +    31 

Coefficient  of  correlation  

.433  - 

b  .015 

Persistency  (kxio8) 

r»  Jw  t^  »  c 
o  o  o  o  o  c 

^ 

- 

^ 

^ 

0 

e 

^ 

>- 

0 

0 

^ 

^f 

o 

e 

o 

o 

^ 

0. 

>- 

^_ 

o 

e 

240     320     400     460     560     640     720     800     680 

Initial  Rate  of  Yield  -Lbs.  f.C.M.  per  Week  (A) 

FIG.  22. — SHOWING  REGRESSION  OF  PERSISTENCY  VALUES  ON  INITIAL  RATE  OF 
YIELD,  HOLSTEIN  RECORDS 

Equation  of  curve:  y  =  3.75  +  .105x. 

k  value  of  the  records  too  low  to  be  representative.  On  the  other  hand, 
since  few,  if  any,  of  the  cows  concerned  were  on  yearly  test,  there  would 
not  be  the  incentive  to  the  owner  to  maintain  milk  flow  at  the  highest 
possible  level  after  the  first  test.  This  condition  would  tend  to  give  a 
higher  value  to  k  than  might  prevail  for  the  same  cows  on  yearly  official 
test.  Other  possible  complicating  factors  have  been  mentioned  above 
(Fig.  7). 

The  regression  of  persistency  on  initial  rate  of  yield  is  given  in 
Fig.  22.  The  regression  equation  derived  from  the  constants  of  Table 
25  is  k  X  103  =  3.75  +  .1054.  It  will  be  seen  from  Fig.  22  that  the 
mean  persistency  values  at  either  extreme  of  the  initial-rate  range  do  not 


402  BULLETIN  No.  288  [April, 

conform  very  well  with  the  regression  curve.  On  the  whole,  however, 
a  linear  description  appears  to  be  justified  for  the  bulk  of  the  records. 
A  noticeable  difference  between  the  Holstein  and  Guernsey  data  is  in 
the  slopes  of  the  curves,  1.782  for  the  Guernsey  and  .735  (=  .105  X  7) 
for  the  Holstein. 

Correction  Factors. — The  initial  rate  of  yield  is  corrected  for  age  to 
the  age  of  maximum  initial  rate  as  per  the  relations  shown  in  Fig.  21. 

A  20-inch  slide  .rule  provided  with  a  specially  graduated  slide  has 
been  used  in  making  these  age  corrections.  The  application  of  the 
method  will  be  readily  apparent.  The  proper  multiplication  factors  at 
certain  specified  ages  are  derived  from  the  equation.  Graduations  are 
made  on  the  slide  corresponding  to  these  factors,  but  labeled  with  the 
appropriate  age  value  instead  of  the  value  of  the  factor  itself.  This 
method  of  computation  is  highly  advantageous  in  point  of  time  and 
convenience.  It  also  permits  rather  fine  age  distinctions  (.1  month) 
at  those  ages,  say  under  three  years,  where  the  factors  are  changing 
most  rapidly.  This  feature  practically  offsets  the  four-figure  limitation 
of  legibility  inherent  in  the  dimension  of  the  rule. 

The  persistency  values  are  corrected  for  initial  rate  of  yield  to  the 
mean  initial  rate,  741,  according  to  Fig.  22,  by  the  equation  kc  =  k0  + 
49.42  —  .105A;  in  which  k0  is  the  observed  value  of  A;  X  103,  and  kc  the 
corresponding  value  corrected  for  initial  rate  of  yield. 

Variability  in  Persistency. — The  distribution  of  the  observed  per- 
sistency values  has  been  given  in  Table  24.  Correction  of  these  values 
for  initial  rate  of  yield  leads  to  the  distribution  given  in  Table  26.  The 
two  distributions  are  presented  graphically  on  a  percentage  basis  in  Fig. 
23. 

The  corrected  persistency  values  give  a  more  regular  distribution, 
a  slightly  higher  mean,  and  considerably  lower  standard  deviation  as 
compared  with  the  raw  values.  The  coefficient  of  variability  is  very 
high  in  either  case.  When  expressed  in  terms  of  the  theoretical  yields, 
the  coefficient  of  variability  takes  a  much  more  usual  value,  namely 
13.05  (see  last  column  of  Table  26).  The  corresponding  figure  for  the 
Guernsey  records  is  14.53.  Hence  it  appears  that  there  is  only  slightly 
less  variability  in  persistency  for  the  Holstein  records  than  that  found 
above  for  the  Guernsey  records. 

Variability  in  Initial  Rate  of  Yield. — The  distribution  with  respect 
to  initial  rate  of  yield  has  been  given  in  Table  24.  Correction  of  these 
values  for  age  gives  the  distribution  presented  in  Table  27.  The  coeffic- 
ient of  variability  is  18.84  and,  as  before  mentioned,  this  value  should  be 
very  closely  related  to  the  same  constant  for  yield  for  the  year.  A  note- 
worthy feature  of  these  Holstein  records  is  the  high  mean  value  of  the 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


403 


theoretical  initial  rate  of  yield,  571.  This  is  equivalent  to  81.6  pounds 
of  4-percent  milk  per  day  for  the  mature  cow,  or  nearly  double  the 
comparable  figure  for  the  Guernsey  records. 


\t> 


14 


Row 


Corrected  k'a 


k'a 


-40 


40  80 

Persistency 


IEO 

(kxio3) 


160 


200 


FIG.    23. — PERCENTAGE    FREQUENCY    DISTRIBUTIONS   FOR 
PERSISTENCY  OF  LACTATION,  HOLSTEIN  RECORDS 

The  corrected  k's  have  been  corrected  for  initial  rate  of 
yield,  as  explained  in  text. 

Rate  of  Yield  and  Yield  for  the  Year. — It  is  of  interest  to  determine 
for  the  Holstein  records  the  point  in  the  lactation  curve  that  affords 
the  best  index  of  yield  for  a  year.  The  records  do  not  give  the  actual 
year's  production,  so  that  a  direct  comparison  such  as  that  made  for  the 
Guernsey  records  is  not  possible.  It  is  possible,  however,  to  consider 
the  problem  on  a  theoretical  basis. 

The  points,  t',  on  the  lactation  curves  at  which  the  area  is  unaffected 
by  variation  in  k  for  the  various  persistency  classes  have  been  given  in 
Table  18.  The  use  of  these  values  in  conjunction  with  the  persistency 
frequency  distribution  of  Table  24  gives  the  following  constants :  mean 


404 


BULLETIN  No.  288 


[April, 


=  5.357  ±  .005;  standard  deviation  =  .291  +  .004;  coefficient  of  vari- 
ability =  5.44  ±  .07.  From  these  results  it  may  be  assumed  that  the 
highest  correlation  between  rate  of  yield  and  yield  for  the  year  would 
be  found  at  about  5.36  months  after  calving.  For  any  single  random 
record  the  chances  are  even  that  the  point  i'  would  lie  within  about  six 

TABLE  26. — VARIABILITY  IN  CORRECTED  PERSISTENCY  VALUES  AND  CALCULATED 
YIELD  FOR  THE  YEAR:  HOLSTEIN  RECORDS 

(The  persistency  values  have  been  corrected  for  initial  rate  of  yield  to  A  =471.) 


Frequency 

k  X  103 
Persistency 
corrected  for 
A 
(Class  mid- 
points) 

Yield  for  year 
A  =471 
(pounds 
F.C.M.) 

2  

-  15 

26  982 

5  

-     5 

25  380 

26  

5 

23  902 

65  

15 

22  537 

132  

25 

21  275 

172  

35 

20  110 

228  

45 

19  028 

233  

55 

18  029 

196  

65 

17  099 

134  

75 

16  237 

100  

85 

15  439 

51  

95 

14  694 

21  

105 

14  000 

11  

115 

13  356 

10  

125 

12  754 

4  

135 

12  194 

1  

145 

11  669 

2  

155 

11  178 

1  

165 

10  721 

0  

175 

0  

185 

1  

195 

9  510 

Mean  

54.23 

18  327 

Standard  deviation  

25.08 

2  392 

Coefficient  of  variability  .... 

46.25 

13.05 

TABLE  27. — VARIABILITY  IN  INITIAL  RATE  OP  YIELD  CORRECTED  FOR  AGE: 
HOLSTEIN  RECORDS 

(Mean  =  571;  standard  deviation  =  107.6;  coefficient  of  variability  =  18.84) 


A«... 

308 

336 

364 

392 

420 

448 

476 

504 

532 

fb  

1 

7 

26 

44 

56 

73 

106 

147 

144 

A.. 

560 

588 

616 

644 

672 

700 

728 

756 

784 

f  

139 

143 

123 

109 

82 

68 

46 

29 

23 

A.. 

812 

840 

868 

896 

924 

952 

980 

1008 

1036 

f  

5 

8 

3 

5 

2 

2 

3 

0 

1 

•Class  mid-points  of  initial  rate  of  yield  in  pounds  of  F.C.M.  per  week. 
bFrequency. 


1927]  PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows  405 

days  of  this  mean  value.  It  is  to  be  expected  on  the  basis  of  the  above 
results  that  the  rate  of  yield  during  the  fifth  month  of  lactation  would 
be  fully  as  highly  correlated  with  the  comparable  yield  for  the  year  in 
the  case  of  these  Holstein  cows  as  has  been  shown  to  be  the  case  for  the 
Guernsey  cows. 

Influence  of  Heredity  and  Environment. — The  comparisons  made  here 
are  similar  to  those  made  above  for  the  Guernsey  records,  with  respect 
to  persistency  and  initial  rate  of  yield.  Data  for  yearly  yield  and  fat 
percentage  for  the  year  being  lacking,  the  comparisons  cannot  be  made 
for  these  items. 

Those  herds  were  selected  in  each  of  which  there  were  at  least  two 
unrelated  cows,  as  above  defined,  and  also  at  least  two  half-sisters  by  a 
common  sire  and  from  different  dams.  The  correlations  with  respect 
to  persistency  and  initial  rate  of  yield,  between  unrelated  cows  on  the 
one  hand  and  between  half-sisters  on  the  other  hand,  have  been  com- 
puted from  the  available  records  of  the  cows  in  these  selected  herds. 

Similar  correlations  have  been  determined  for  half-sisters  from  a 
common  dam  and  by  different  sires;  for  full  sisters;  for  dam  and  daughter; 
and  for  different  lactation  records  of  the  same  cow.  All  herds  which  pro- 
vided two  or  more  records  of  any  one  of  the  relationships  specified  in  this 
paragraph  have  been  used  in  order  to  secure  as  large  numbers  as  possible. 

It  may  be  assumed  that  any  correlation  between  unrelated  cows  is 
due  to  the  environmental  relationship  of  being  in  the  same  herd.  For 
the  other  groups  there  is  added  to  this  environmental  relationship  the 
blood  relationship  specified  by  the  basis  of  the  selection  and,  for  the  one 
case,  the  identity  of  the  animal. 

The  number  of  herds  used  in  the  comparisons  for  unrelated  cows 
and  half-sisters  by  a  common  sire  is  55;  the  number  of  combinations  of 
unrelated  cows,  1,620;  the  number  of  combinations  of  half-sisters  by  a 
common  sire,  1,774;  and  the  number  of  such  common  sires,  126.  The 
number  of  combinations  of  half-sisters  from  a  common  dam  is  78  and  of 
full  sisters,  64.  There  are  83  dam-daughter  pairs,  and  96  cows  having 
two  lactation  records.  The  various  statistical  constants  are  given  in 
Table  28. 

The  mean  values  of  Table  28  show  a  general  agreement  between  the 
several  groups,  altho  with  some  points  of  diversity.  It  will  be  noted 
that  the  half-sister  (common  sire)  herd  mates  and  the  unrelated  herd 
mates  agree  quite  closely  in  age-corrected  initial  rate  of  yield  and  also 
in  persistency  corrected  for  initial  rate  of  yield.  The  daughters  of  the 
dam-daughter  group  show  an  exceptionally  high  rate  of  decrease  in  yield. 
The  constants  of  the  lactation  curves  indicate  that  the  daughters  of  the 
dam-daughter  group  would  have  a  somewhat  lower  yearly  yield  than 
their  dams. 


406 


BULLETIN  No.  288 


[April, 


The  initial  rate  of  yield  of  the  second  records  of  the  96  cows  in 
the  last  group  show  an  increase  of  48.8  pounds  F.C.M.  per  week  over  the 
first  records.  The  persistency  values  are  practically  the  same.  Evi- 
dently the  correction  factors  have  served  to  correct  the  indirect  influence 


TABLE  28. — STATISTICAL  CONSTANTS  OF  CERTAIN  CHARACTERS  WITHIN  GROUPS 

SELECTED  ON  THE  BASIS  OF  BLOOD  AND  ENVIRONMENTAL  RELATIONSHIP: 

HOLSTEIN  RECORDS 


Intragroupal  relationship 

k  X  103 
Persistency 
corrected  for 
A 

Theoretical 
initial  rate  of 
yield  corrected 
for  age  (pounds 
F.C.M.  per 
week) 

Means 

Unrelated  herd  mates8  

55.31  +     .43 
56.29  +     .40 
50.58  +  1.04 
45.38  +  1.42 
49.57  +  1.71 
59.73  +  2.16 
48.45  +  1.36 
48.73  ±  1.77 

574.6  +  1.7 
580.4  ±  1.8 
609.3  ±7.5 
582.2  +  9.7 
585.4  +  7.0 
585.2  +  8.2 
569.8  ±  7.1 
618.6  ±  7.2 

Half-sister  herd  mates  (C-S)b  

Half-sister  herd  mates  (C-D)C   

Full-sister  herd  mates  

T-»        t       i  ,     i_     j             /  dams.  . 

Dam-daughter  herd  mates  |  daughterg 

i      jj          f  first  records.  . 

Same  cow,  same  herd*.  .  .  .  {  second  recordg  

Standard  deviations 

Unrelated  herd  mates  

25.74+     .31 
25.22±     .29 
13.57  +     .73 
16.89  +  1.01 
23.12  +  1.21 
29.14  +  1.53 
19.74+     .96 
25.65  +  1.25 

99.9  +  1.2 
113.0  +  1.3 
97.6  +  5.3 
115.6  ±  6.9 
94.3  ±4.9 
111.4  ±  5.8 
102.8  +  5.0 
105.1  ±  5.1 

Half-sister  herd  mates  (C-S)  

Half-sister  herd  mates  (C-D)  

Full-sister  herd  mates  

T-I       j  •  '•    i_     i_     j              /  dams.  . 

Dam-daugher  herd  mates.  (  daughters     

a                          i      j           1  first  records.  . 

Same  cow,  same  herd.  .  .  .    {  gecond  records  

Coefficients  of  variability 

Unrelated  herd  mates  

46.53  ±     .55 
44.81  +     .51 
26.83  +  1.45 
37.22  +  2.22 
46.64  +  2.44 
48.79  +  2.55 
40.74  +  1.98 
52.63  ±  2.56 

17.38+     .21 
19.48  +     .22 
16.02  +     .87 
19.86  +  1.18 
16.10  +     .84 
19.04  ±  1.00 
18.05  +     .88 
16.99  +     .83 

Half-sister  herd  mates  (C-S)  

Half-sister  herd  mates  (C-D)  

Full-sister  herd  mates  

Dam-daughter  herd  mates  /  dams  

\  daughters  

Same  cow,  same  herd  ....    f  first  records  

1  second  records  

Coefficients  of  correlation 

Unrelated  herd  mates  

.144  +  .016 
.239  ±  .015 
.478  +  .059 
.067  +  .084 
.462  +  .058 
.461  +  .054 

.064  ±  .017 
.336  ±  .014 
.261  ±  .071 
.367  ±  .073 
.329  ±  .066 
.438  +  .056 

Half-sister  herd  mates  (C-S)  

Half-sister  herd  mates  (C-D)  

Full-sister  herd  mates  

Dam-daughter  herd  mates  

Same  cow,  same  herd  

•Not  related  as  dam  and  daughter,  full  sister  or  half-sister.  bCommon  sire, 
different  dams.  cCommon  dam,  different  sires.  Successive  lactation  records  of  the 
same  cow. 


1927} 


PERSISTENCY  OP  LACTATION  IN  DAIRY  Cows 


407 


of  advance  in  age  on  persistency,  but  have  not  fully  corrected  the  more 
direct  influence  of  advance  in  age  on  initial  rate  of  yield.  Expressed  in 
terms  of  yearly  yield,  the  second  record  shows  an  increase  in  yield  in 
excess  of  that  expected  on  the  basis  of  the  average  change  in  yield  with 
age  for  the  entire  number  (1,395)  of  cows.  The  excess  increase  amounts 
to  about  8-percent  of  the  first  record.  The  most  likely  explanation  for 
this  excess  increase  is  the  artificial  factor  that  there  is  no  commercial 
object  in  conducting  and  reporting  a  second  test  on  a  cow  unless  that 
second  test  shows  a  relatively  higher  production  than  the  first  test. 


TABLE  29. — ESTIMATED  COEFFICIENTS  OF  CORRELATION  WITH  RESPECT  TO  PERSIS- 
TENCY AND  INITIAL  RATE  OF  YIELD,  BETWEEN  Cows  OF  VARIOUS  BLOOD 
RELATIONSHIPS  WITHIN  AN  INDEFINITELY  LARGE  HERD: 
HOLSTEIN  RECORDS 


Relationship 

Characters 

Persistency 
of  lactation 
corrected  for 
A 

Initial  rate 
of  yield  cor- 
rected for 
age 

Half  -sister,  common  sire  

.111  +  .016 
.390  ±  .065 
-  .090  +  .084 
.371  +  .064 
.370  +  .059 

.290  +  .015 
.211  +  .073 
.324  +  .075 
.284  +  .068 
.400  ±  .058 

Half-sister,  common  dam  

Full-sister  

Dam-daughter  

Same  cow"  

•Correlation  between  successive  lactation  records  of  the  same  cow. 


There  is  also  the  possibility  of  the  excess  increase  being  due  to  a  physi- 
ological factor  in  the  nature  of  an  increased  development  of  the  mam- 
mary functions  due  to  the  exercise  and  training  of  the  first  test.  The 
latter  explanation  has  been  noted  by  Gowen9  and  emphasized  by  Graves 
and  Fohrman10. 

The  coefficients  of  correlation  show  an  appreciable  resemblance 
between  unrelated  herd  mates  with  respect  to  persistency  and  a  smaller 
degree  of  resemblance  with  respect  to  initial  rate  of  yield.  The  cor- 
relations for  the  various  relationships  within  a  single  herd  estimated  by 
the  method  used  above  are  given  in  Table  29. 

These  correlations  with  environment  constant  indicate  a  very  slight 
relation  between  half-sisters  by  a  common  sire  with  respect  to  per- 
sistency of  lactation,  namely,  r  =  .111.  But  the  relation  between  half- 
sisters  from  a  common  dam  is  much  more  marked,  r  =  .390.  The  re- 
lation between  dam  and  daughter  is  of  the  same  order,  r  =  .371.  And 
both  these  are  slightly  more  closely  related  than  are  the  first  and  second 
records  of  the  same  cow,  r  =  .370. 


408  BULLETIN  No.  288  [April, 

The  resemblance  in  persistency  of  cows  related  thru  the  dam 
appears  to  be  much  greater  than  those  related  thru  the  sire.  This  is  in 
agreement  with  the  Guernsey  data.  Also,  like  the  Guernsey,  the  Hoi- 
stein  full  sisters  show  no  significant  correlation  in  persistency.  It  is 
difficult  to  reconcile  the  results  for  full  sisters  with  the  results  for  other 
relationships,  for  full  sisters  have  a  common  dam  and  it  would  seem 
should  show  as  close  a  resemblance  as  half-sistefs  from  a  common  dam. 

While  the  small  numbers  reduce  the  significance  of  the  correla- 
tions as  a  basis  of  generalization,  the  general  agreement  of  the  Guernsey 
and  Holstein  data  make  it  appear  that  there  is  the  possibility  of  a  real 
difference  in  the  correlation  between  the  persistency  values  of  cows 
according  to  their  relationship  thru  the  sire  or  thru  the  dam. 


DISCUSSION 

Selection  of  Records. — Certain  of  the  Guernsey  records,  taken  in 
the  order  of  publication,  have  been  excluded  in  the  results  as  presented 
above.  Only  original  entries  have  been  used.  A  further  selection  was 
made  on  the  basis  of  time  of  breeding,  so  as  to  eliminate  the  complica- 
tion of  the  effect  of  advanced  pregnancy  on  the  lactation  curve.  The 
records  showing  very  irregular  lactation  curves  were  also  excluded.  It 
is  desirable  to  examine  these  records  excluded  in  the  above  results. 

An  indication  of  the  effect  of  the  selection  on  the  basis  of  time  of 
breeding  is  afforded  by  the  graphic  presentation  of  Fig.  24.  The  data 
on  which  this  figure  is  based  are  taken  from  Bulletin  2726  and  include 
reentry  records.  There  is  considerable  variability  in  the  average  initial 
rate  of  yield  of  the  several  groups  separated  according  to  the  time  of 
breeding  (the  farrow  cows  are  shown  as  bred  12^  months  after  calving). 
There  is  a  very  slight  tendency  to  a  higher  initial  rate  of  yield  in  the 
groups  in  which  breeding  is  delayed.  On  the  other  hand,  there  is  quite  a 
marked  tendency  to  a  slower  rate  of  decrease  in  yield  with  the  delayed 
breeding.  The  basis  on  which  this  rate  of  decrease  has  been  determined 
practically  eliminates  the  effect  of  pregnancy. 

Delayed  breeding  is  sometimes  practiced  to  secure  as  large  a  record 
for  the  year  as  possible.  The  relations  shown  in  Fig.  24  may  be  taken  to 
mean  that  there  is  no  particular  tendency  to  select  inherently  higher 
yielding  cows  to  be  held  open,  but  that  there  is  a  tendency  to  so  manage 
the  groups  in  which  breeding  is  delayed  as  to  cause  them  to  hold  up  in 
milk  flow  better  than  the  average.  On  this  account  the  exclusion  of 
records  with  a  service  period  of  less  than  six  months  tends  to  disturb  the 
persistency  distribution  curve  and  to  give  a  yield  for  the  year  somewhat 
higher  than  would  hold  for  all  the  original  entries. 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


409 


The  records  which  were  excluded  on  account  of  the  irregular  nature 
of  the  lactation  curve  are  treated  in  Fig.  25.  The  four  individual  lacta- 
tion curves  given  serve  to  illustrate  the  diversity  found  as  between 
individual  records.  The  curve  for  No.  6356  shows  an  increasing  rate  of 
yield  for  five  months,  followed  by  a  decreasing  rate  of  yield;  the  curve 
for  No.  7405  is  roughly  the  inverse  of  that  for  No.  6356.  The  curve  for 
No.  8032  shows  a  high  rate  of  decline  in  approximate  conformity  with  the 
equation  type  for  six  months,  after  which  the  rate  of  yield  holds  nearly 
constant;  the  curve  for  No.  8777  on  the  other  hand,  shows  an  increasing 
rate  of  decline  during  the  last  six  months. 


57 
55 
53 
•51 
49 


29 


Number  of  Cows 

13£  255  335  746   754575  572  Z£, 


235   172    196   £55 


Initial  Rate 
Persistency 


1200 

1160 


1120 
HOG 


1080  o 
1060  £ 


1        a       3      4       5       fe       7       8       9       10      11'      ia     13 

Service  Period -Months 

FIG.    24. — SHOWING   THE    RELATION    BETWEEN    SERVICE    PERIOD    AND 
INITIAL  RATE  OP  YIELD  AND  PERSISTENCY,  GUERNSEY  RECORDS 

Accepting  the  exponential  curve  as  representing  the  normal  course 
of  affairs,  it  is  apparent  that  the  normal  course  is  altered  markedly  in 
individual  cases.  When  the  individual  curves  of  the  irregular  records 
(142)  are  thrown  together  and  treated  as  a  group,  they  fall  in  a  fairly 
regular  order  (the  circles  in  Fig.  25).  Judged  by  the  average  curve  alone, 
these  irregular  records  conform  very  well  to  the  exponential  equation. 
The  deviations  of  individuals  are  therefore  of  a  compensating  nature. 

Chemical  Interpretation  of  the  Lactation  Curve* — Brody  et  al,1  on 

"Since  this  material  was  prepared,  a  somewhat  more  extensive  treatment  has 
been  published  by  the  author  in  an  article,  "Interpretation  of  the  Lactation  Curve," 
Jour.  Gen.  Physiol  10,  No.  1,  27-31.  1926. 


410 


BULLETIN  No.  288 


[April, 


the  basis  of  the  equivalence  of  the  equations  of  the  lactation  curve  and 
that  of  a  monomolecular  reaction,  suggest  that  the  rate  of  milk  secretion 
is  limited  by  such  a  chemical  reaction.  The  heterogeneous  nature  of  the 
performance  of  individual  cows  naturally  raises  some  question  as  to  the 
validity  of  such  an  interpretation. 


45 


40 


I 

cu 
n. 


35 


o 
III 

<0 


20 


15 


468 

Morrths  after  Calving 


10 


FIG.  25. — IRREGULAR  LACTATION  CURVES,  GUERNSEY  RECORDS 

Four  individual  lactation  curves  (A.R.  Nos.  6356,  7405,  8032,  and  8777)  are 
shown  to  illustrate  the  variation  found  in  individual  cases  among  the  records  ex- 
cluded on  the  ground  of  the  irregular  nature  of  the  lactation  curves.  The  circles  repre- 
sent the  average  of  the  142  records  thus  excluded.  Equation  of  the  corresponding 
smooth  curve:  y  =  35.78e  — 03685(. 

On  the  basis  of  such  a  chemical  interpretation  the  rate  of  decline 
in  milk  yield  is  governed  by  the  velocity  constant,  k  of  the  equation. 
Tables  8  and  26  show  a  very  high  degree  of  variability  in  the  constant  k. 
It  may  be  questioned  whether  the  velocity  constant  of  a  particular 
chemical  reaction,  under  constant  temperature  conditions,  would  show 
such  a  high  degree  of  variability.  The  conception  of  a  limiting  reaction 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


411 


of  this  kind  furthermore  is  complicated  by  the  fact  that  5  percent  of  the 
individual  records  show  an  increasing  rate  of  yield  with  advance  in  lacta- 
tion for  ten  to  twelve  months.  Apparently,  if  it  is  to  be  postulated  that 
the  rate  of  milk  secretion  is  determined  by  a  limiting  substance,  it  must 
be  further  postulated  that  this  limiting  substance  is  not  all  present  or 
active  until  some  time  (up  to  12  months)  after  the  beginning  of  lactation. 
Breed  Lactation  Curves. — The  initial-rate-of-yield  distributions  shown 
by  the  Guernsey  records  (Table  9)  and  by  the  Holstein  records  (Table 
27)  are  brought  together  on  a  percentage  basis  in  Fig.  26.  There  is 
apparent  a  very  distinct  difference  in  the  two  breeds  with  respect  to  the 


if2 

ih 

<D 
0)6 

1 

8 

&4r 


CSuernsev 


hofeteir 


20 


36 


5Z  68  84  100  116 

Initial  Rate  of  Yield  -Lbs  f.C.M.  per  Pay 


132 


we 


FIG.   26. — SHOWING  PERCENTAGE  FREQUENCY'  DISTRIBUTION  CURVES 

OF  THEORETICAL  INITIAL  RATE  OF  YIELD  FOR  GUERNSEY  AND 

HOLSTEIN  RECORDS 

The  data  of  both  curves  have  been  corrected  for  age  to  the  age  of 
maximum  initial  rate  of  yield. 


A  constant  of  the  lactation  curves.  So  far  as  indicated  by  the  theoretical 
initial  rate  of  yield,  the  Holstein  records  show  a  much  higher  productive 
capacity. 

The  persistency  frequency  distributions  of  the  two  breeds  are  shown 
in  Fig.  27.  The  two  curves  in  this  instance  are  more  nearly  coincident  but 
are  not  so  directly  comparable.  Taken  as  they  stand,  they  indicate  that 
the  Holstein  records  show  a  more  rapid  rate  of  decrease  in  rate  of  yield 
with  advance  in  lactation  than  do  the  Guernsey  records.  The  element 
that  disturbs  such  a  direct  comparison  is  the  fact  that  the  persistency 
values  have  been  corrected  to  a  different  initial  rate  base  in  the  two  cases. 

The  mean  persistency  value  of  the  Guernsey  records  corrected  to  the 
mean  initial  rate  is  k  X  103  =  44.81  (Table  8).  If  correction  were  made 


412 


BULLETIN  No.  288 


[April, 


to  the  mean  initial  rate  of  the  Holstein  records  by  the  equation  for  the 
Guernsey  records,  the  mean  persistency  value  for  the  Guernsey  records 
would  become  k  X  103  =  94.90,  as  compared  with  k  X  103  =  54.23  for 
the  Holstein  records. 

The  mean  persistency  value  of  the  Holstein  records  corrected  to  the 
mean  initial  rate  is  k  X  103  =  54.23  (Table  26).  If  correction  were 
made  to  the  mean  initia.1  rate  of  the  Guernsey  records  by  the  equation  of 
the  Holstein  records,  the  mean  persistency  value  for  the  Holstein  records 
would  become  k  X  103  =  33.62,  as  compared  with  k  X  103  =  44.81 
for  the  Guernsey  records. 


entage 

o> 


Guorns 


Hpfctelr 


-40 


4O  8O  I2O  160 

Persistency   (kxios) 


200 


FIG.    27. — SHOWING    PERCENTAGE    FREQUENCY    DIS- 
TRIBUTION CURVES  OF  PERSISTENCY  OF  LACTA- 
TION FOR  GUERNSEY  AND  HOLSTEIN  RECORDS 

The  persistency  values  refer  to  the  rate  of  decrease 
per  mille  per  month  in  the  rate  of  yield  of  4  percent 
milk  per  month.  The  data  of  both  curves  have  been 
corrected  for  initial  rate  of  yield  to  the  mean  initial  rate 
of  yield:  namely,  39.2  pounds  F.C.M.  per  day  for  the 
Guernsey  records  and  67.3  pounds  F.C.M.  per  day  for 
the  Holstein  records. 


Hence  it  appears  that  while  the  Holstein  records  show  a  greater 
rate  of  decrease  than  the  Guernsey  records,  yet  when  initial  rate  of  yield 
is  allowed  for,  the  reverse  is  true.  In  view  of  the  difference  of  the  slope 
of  the  regression  lines  of  persistency  on  initial  rate  of  yield  (Fig.  28),  it 
is  a  question  as  to  just  how  the  two  breed  records  may  equitably  be  com- 
pared with  respect  to  persistency,  altho  it  would  seem  proper  to  say 
that  the  Holstein  records  show  greater  persistency  (smaller  value  of  k) 
than  the  Guernsey  records. 

Ellinger3  mentions  that  the  Red  Danish  breed  is  more  persistent 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


413 


than  the  Jersey  breed.  His  measure  of  persistency  is  the  ratio  of  the 
milk  yield  of  the  second  10-week  period  of  the  lactation  to  the  milk  yield 
of  the  first  10-week  period.  Possibly  the  breed  differences  in  persistency 
are  associated  with  size  (weight),  the  larger  cow  tending  to  be  more  per- 
sistent than  the  smaller  cow. 

Carlyle  and  Woll2  report  results  on  persistency  for  cows  in  the 
experiment  station  herd  classified  in  three  groups:  A,  extreme  dairy 
type  (9  Jerseys,  4  Guernseys,  4  Holsteins) ;  B,  large  dairy  type  (3  Jerseys, 
5  Guernseys,  4  Holsteins);  C,  dual-purpose  type  (11  Shorthorns,  1  Red 


120 


40 


Zi  48  £4  80  96  |l£ 

Initial  Rate  of  Yield  -Lbs.  f.  C.M.  per  Pay 


126 


FIG.  28. — SHOWING  THE  REGRESSION  OF  PERSISTENCY  OF  LAC- 
TATION ON  INITIAL  RATE  OF  YIELD  FOR  GUERNSEY 
AND  HOLSTEIN  RECORDS 

Polled).  Their  measure  of  persistency  is  the  decrease  in  milk  or  fat 
yield  of  a  late  week  in  lactation  as  compared  with  an  early  week,  the 
decrease  being  expressed  as  a  percentage  of  the  yield  of  the  early  week. 
Their  results  are  somewhat  variable,  according  to  the  particular  two 
weeks  of  the  lactation  compared,  but  in  general  the  extreme  dairy  type 
showed  the  smallest  decrease,  the  large  dairy  type  the  largest  decrease, 
and  the  dual-purpose  type  intermediate.  These  results  are  of  interest 
in  connection  with  the  prevalent  opinion  as  to  the  excellence  of  the  dairy 
breeds  in  persistency  of  lactation,  as  compared  with  the  dual-purpose 
breeds.  It  should  be  pointed  out  that  the  Wisconsin  herd  was  probably 
composed  of  more  or  less  select  individuals  and  the  same  results  might 
not  hold  for  a  larger  and  more  representative  population. 

The  breeding  experiments  of  the  Iowa  Station  are  of  interest  at 
this  point.  The  experiments  referred  to  are  based  on  the  mating  of 
dairy  bred  bulls  with  scrub  cows.  McCandlish  et  aln  have  studied  the 
persistency  of  lactation  of  scrub  cows,  of  purebred  dairy  cows,  and  of 
crossbred  cows  by  dairy  bulls  from  the  scrub  cows.  They  have  measured 
persistency  by  expressing  each  month 's  milk  yield  as  a  percentage  of  the 
first  month's  milk  yield.  As  thus  expressed,  the  scrubs  decrease  most 


414  BULLETIN  No.  288  [April, 

rapidly,  the  dairy  bred  cows  least  rapidly,  and  the  crossbreds  are  inter- 
mediate between  the  parent  types.  These  results  would  indicate  that 
there  is  a  difference  between  dairybred  cows  and  unimproved  cows  in 
respect  to  persistency  of  lactation.  Correction  of  the  persistency  values 
for  initial  rate  of  yield  would  likely  make  the  difference  more  pronounced. 
Apparently  also  the  sire  and  dam  influence  the  persistency  of  the  off- 
spring about  equally. 

Measures  of  Persistency? — The  measures  of  persistency  mentioned 
above,  it  will  be  noted,  are  all  based  on  a  ratio  of  one  sort  or  another.  To 

total  lactation  yield 

these  should  be  added  the  measure  of  Sanders/6      — ; —    — - — ,'         •• 

maximum  day  s  yield 

The  length  of  the  lactation  period,  and  consequently  the  lactation  yield, 
are  greatly  affected  by  the  length  of  the  service  period.  Sanders  has 
therefore  applied  a  correction  factor  to  the  ratios  to  reduce  them  to  a 
standard  service  period.  If  yield  for  a  definite  time  period  were  sub- 
stituted for  lactation  yield  in  the  ratio  of  Sanders,  it  is  apparent  the 
ratio  would  acquire  a  definite  relation  to  the  rate  of  decrease  in  yield. 
Turner20  has  presented  a  table  for  converting  such  ratios  into  the  cor- 
responding percentage  decrease  per  month.  It  would  seem  preferable, 
in  the  use  of  such  a  system,  to  substitute  a  smoothed  value  of  an  initial 
period  yield  in  place  of  the  maximum  yield.  It  is  evident  that  the  system 
applied  literally  would  not  distinguish  between  the  performance  of  those 
cows  which  show  an  increasing  rate  of  yield  and  those  which  show  an 
approximately  equal  decreasing  rate  of  yield. 

Turner,  in  the  paper  mentioned,  proposes  also  the  method  of 
dividing  the  yield  for  each  calendar  month  by  that  of  the  preceding  calen- 
dar month  and  using  the  arithmetic  average  of  the  ratios  thus  secured 
as  a  measure  of  persistency.  It  should  be  noted  that  such  a  method  is 
mathematically  unsound  for  the  purpose  in  view,  and  where  applied  to 
the  irregular  data  of  individual  records,  tends  to  lead  to  too  high  results. 

Correction  Factors  for  Length  of  Record. — A  set  of  correction  factors 
for  length  of  record  from  200  to  365  days  was  presented  in  Bulletin 
272,6  based  on  an  average  persistency  value  for  Guernsey  records  of 
k  X  103  =  44.12.  It  was  pointed  out  that  the  factors  would  vary 
according  to  the  values  of  k.  The  k  frequency  distribution  (Fig.  14  and 
Table  6)  gives  an  indication  of  the  variability  of  k.  It  is  apparent  from 
the  relation  between  k  and  yield  (r  =  —  .226)  that  the  correction  factors 
given  are  based  on  too  high  a  value  of  k  for  low  yields  and  too  low  a 
value  of  k  for  high  yields.  There  is  no  simple  way  of  taking  account  of 

"Since  this  material  was  prepared,  a  somewhat  more  extensive  treatment  has 
been  given  by  the  author  in  a  paper  submitted  to  the  Journal  of  Agricultural  Re- 
search entitled,  "Measures  of  Persistency  of  Lactation." 


1927]  PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows  415 

this  known  relation,  but  as  the  factors  stand  the  error  to  which  they  are 
subject  is  not  very  serious  so  far  as  application  to  the  Guernsey  Ad- 
vanced Registry  records  is  concerned.  It  appears  further  that  they 
should  apply  almost  equally  well  to  Holstein  records,  so  far  as  may  be 
judged  by  the  present  Holstein  data. 

Persistency  as  a  Heritable  Character. — Sanders,16  from  a  study  of  the 
records  of  Shorthorn  cows  in  an  English  milk  recording  society,  came 
to  the  conclusion  that  the  shape  of  the  lactation  curve,  tho  largely  de- 
termined by  environmental  factors,  is  due  partly  to  a  genetic  character- 
istic of  the  cow.  His  evidence  for  the  genetic  basis  of  persistency  is  the 

total 

relation  between  the  standard  deviation  of  the  -  -  ratios  of  in- 

maximum 

dividual  records  as  compared  with  the  standard  deviation  of  individual 
mean  ratios. 

It  is  obvious  that  environment  may  be  a  large  factor  in  determining 
persistency  of  lactation.  Lactation  may  be  terminated  quickly  at  any 
stage  by  failure  to  remove  the  accumulated  milk  from  the  udder.  It 
seems  possible,  therefore,  that  the  character  and  frequency  of  milking 
may  be  factors  in  persistency.  Obviously  the  feed  supply  is  a  very  im- 
portant factor  and  probably  largely  determines  the  difference  between 
advanced  registry  practice,  which  results  in  an  average  rate  of  decrease 
of  about  5  percent  per  month,  and  commercial  practice,  which  results 
in  an  average  rate  of  decrease  of  about  10  percent  per  month. 

The  data  of  McCandlish  et  alu  presumably  were  obtained  with  en- 
vironment constant,  or  nearly  so,  that  is,  within  the  same  herd.  The 
results  seem  to  show  unequivocally  that  dairy-bred  cows  are  superior 
to  scrub  cows  in  persistency  of  lactation,  and  that  the  character  is 
inherited  apparently  in  a  blending  fashion.  It  may  be  noted  incidental- 
ly that  the  lactation  curve  of  the  dairy-bred  cows  shows  a  rate  of  decrease 
of  about  10.6  percent  per  month. 

The  results  of  the  present  paper  indicate  that  persistency  of  lact- 
ation is  nearly  as  definite  an  individual  character  as  is  initial  rate  of  yield, 
judged  by  the  fact  that  with  the  herd  constant  the  correlation  between 
two  lactation  records  of  the  same  cow  is  r  =  .370  for  persistency  and  r  = 
.400  for  initial  rate  (Table  29).  Neither  of  these  coefficients  is  as  high  as 
similar  comparisons  reported  by  Gowen8  for  the  Holstein  breed  in  the 
case  of  yearly  milk  yield  (r  =  .667)  or  fat  percentage  (r  =  .715).  The 
effect  of  environment  is  not  allowed  for  in  the  results  of  Gowen  just 
quoted. 

It  is  very  clear  that  there  is  considerable  variability  between  the 
individual  records  of  cows  with  respect  to  persistency  and,  admitting 
that  this  is  in  part  due  to  genetic  differences  of  the  individual  cows  con- 


416  BULLETIN  No.  288  [April, 

cerned,  it  follows  that  either  the  sire  or  the  dam,  or  both,  must  have  an 
influence  on  the  persistency  of  the  daughter.  In  the  case  of  the  Guernsey 
records  it  seems  clear  that  as  between  the  97  sires  studied  there  was  no 
difference  in  their  effect  on  the  persistency  of  their  daughters.  Either 
the  sire  has  no  influence  on  the  persistency  of  the  daughter,  or  the  97 
sires  were  all  genetically  alike  with  respect  to  this  character.  The  latter 
alternative  is  forced  by  the  results  of  McCandlish  referred  to  above. 
While  the  Holstein  records  show  a  statistically  significant  correlation  be- 
tween half-sisters  by  a  common  sire  and  from  different  dams,  the  coeffic- 
ient itself  (r  =  .111  ±  .016)  is  too  low  to  be  of  much  practical  importance. 
Obviously,  if  the  differences  in  sires  are  not  responsible  for  the  inherent 
qualities  of  the  daughters  with  respect  to  persistency,  then  differences 
in  the  dams  must  be  responsible.  Half-sisters  from  a  common  dam  and 
by  different  sires,  as  well  as  daughter  and  dam,  show  a  significant  and 
material  correlation,  which  is  in  accord  with  such  an  elimination  con- 
clusion. The  anomalous  results  for  full  sisters  on  the  other  hand  raise 
the  question  as  to  whether  there  is  really  any  genetic  difference  between 
individuals  within  either  of  the  two  breeds  with  respect  to  persistency  of 
lactation.  That  is  to  say,  it  may  be  possible  that  there  is  very  little 
variability  of  a  genetic  nature  among  our  present  dairy  breeds  with  re- 
spect to  persistency  of  lactation. 

The  Short-Time  Test. — Gavin7  was  perhaps  the  first  to  show  definite- 
ly, by  statistical  treatment,  the  correlation  between  milk  yield  for  a 
short  period  at  the  start  of  lactation  and  milk  yield  for  the  entire  lacta- 
tion. He  found  the  correlation  between  lactation  yield  and  yield  for  the 
5th  to  12th  weeks  to  be  r  =  .858  +  .005,  in  the  case  of  normal  lactations 
35  to  45  weeks  in  length.  The  corresponding  relation  for  the  maximum- 
days'  yield  was  r  =  .839  +  .006.  The  first  correlation  corresponds 
more  or  less  closely  in  nature  with  the  present  correlation  between  the 
observed  rate  of  yield  two  months  after  calving  and  yield  for  the  year, 
shown  in  Table  17  to  be  r  =  .827  ±  .006.  Gavin's  data  were  taken  from 
the  herd  records  of  Lord  Rayleigh  in  England.  It  is  evident  from  the 
coefficients  of  correlation  that  the  present  Guernsey  Advanced  Regis- 
try data  are  very  similar  to  these  English  herd  records  with  respect  to 
the  closeness  of  the  relation  under  consideration. 

Reference  to  Table  17  and  Fig.  18  shows  further  that  the  correlation 
between  rate  of  yield  and  yield  for  the  year  increases  to  r  =  .935  +  .002 
at  six  months  after  calving.  This  correlation  would  naturally  be  higher 
except  for  deviations  of  the  observed  lactation  curve  from  the  smoothed 
curve.  The  relation  between  the  smoothed  rate  of  yield  and  yearly 
yield  is  shown  in  Table  17  to  be  r  =  .980  ±  .001. 

A  question  arises  as  to  the  records  rejected  because  of  unusual 


1927]  PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows  417 

irregularities.  The  correlation  between  the  yield  of  the  sixth  month 
and  the  yearly  yield  for  these  142  irregular  records  works  out  at  r  =  .839 
±  .017,  or  materially  lower  than  that  found  for  the  bulk  of  the  records. 
Inclusion  of  these  142  irregular  records  with  the  other  1,534  records 
gives  a  correlation  of  r  =  .928  ±  .002. 

Since  the  rate  of  yield  is  so  closely  related  to  the  yearly  yield  it  is  a 
question  why,  in  the  interest  of  economy,  a  short-time  test  should  not 
be  used.  A  short-time  test  conducted  in  the  fifth  month  of  lactation 
affords  a  well  nigh  perfect  index  of  the  cow 's  365-day  energy  yield  under 
the  conditions  of  the  Guernsey  Advanced  Registry  where  pregnancy 
during  the  year  period  is  eliminated  as  a  factor.  If  the  365-day  yield  is 
the  essential  measure  of  a  cow's  performance,  as  seems  to  be  assumed  by 
the  practice  of  the  advanced  registry  system,  why  not  use  a  six-months- 
after-calving  short-time  test  to  measure  the  cow 's  performance? 

But  the  question  should  be  asked,  why  select  365  days  as  a  proper 
length  of  time  to  test  a  cow  when  that  is  only  a  part  of  her  lactation 
period  under  the  conditions  that  have  developed  in  official  testing? 
The  selection  of  the  365-day  period  is  undoubtedly  connected  with  the 
economic  significance  of  the  year's  yield  under  ordinary  conditions  of 
milk  production,  when  the  cows  are  bred  to  freshen  once  a  year.  The 
feed  cost  of  milk  production  under  such  conditions  bears  a  very  definite 
relation  to  the  yearly  yield. 

The  relation  depends  upon  the  two  main  physiological  facts  that 
a  certain  quantity  of  nutrients  is  required  for  body  maintenance  and  a 
certain  further  quantity  is  required  for  the  elaboration  of  the  milk  itself. 
For  the  sake  of  simplicity  we  may  consider  a  cow  of  1,100  pounds  live 
weight.  According  to  Haecker,11  the  maintenance  of  an  1100-pound 
cow  for  one  year  requires  3,182  pounds  of  digestible  nutrients,  and  the 
elaboration  of  one  pound  F.C.M.  (4-percent  milk)  requires  .343  pounds 
of  digestible  nutrients  (cf.  Gaines4).  These  quantitative  relations  lead 

3182 

to  the  equation  y  =  .343  H — ,  where  y  is  the  feed  cost  in  pounds  of 

x 

nutrients  per  pound  F.C.M.  and  x  is  the  yearly  yield  in  pounds  F.C.M. 
The  curve  of  this  equation  is  given  in  Fig.  29.  Cost  in  nutrients  may  be 
translated  directly  into  cost  in  dollars,  as  has  been  shown  by  Ross  et  al.lb 

Fig.  29  shows  clearly  how  the  yearly  yield  is  related  to  the  cost  of 
production,  and  why  high  yearly  production  is  so  important  from  the 
standpoint  of  efficiency  of  production.  It  should  be  emphasized  that  the 
relation  between  yield  and  cost  as  a  generalization  is  based  on  a  calving 
interval  (time  from  one  calving  to  the  next  calving)  of  one  year. 

The  advanced  registry  record  is  often  crowded  past  the  point  "X" 
of  maximum  efficiency  of  Fig.  29  and,  also,  usually  fails  to  meet  the  calv- 


418 


BULLETIN  No.  288 


[April, 


ing  interval  condition  mentioned  above.  While  the  selection  of  a  year 
period  as  the  time  over  which  production  is  to  be  measured  has  a  very 
fundamental  basis  in  ordinary  practice,  the  365-day  official  record  has 


1.4 


1.2 

2 

o 
ttl  1.0 


<0 

•E 

0) 

I 


.2 


£0  40  <40  60  100 

F-  C.  M.  per  Year  -  Cwt. 


ISO 


140 


FIG.  29. — THEORETICAL  RELATION  OF  YEARLY  YIELD  TO 
EFFICIENCY  OF  PRODUCTION 

The  solid  curve  represents  the  physiological  limitation  of  efficiency  of  milk 
production  by  the  1100-pound  cow,  according  to  Haecker's  feeding  standard.  The 
asterisk  (*)  on  the  curve  shows  the  point  at  which  is  reached  the  often  quoted 
efficiency  of  the  cow,  that  is  18  percent  of  the  feed  nutrients  recovered  in  the  milk. 
The  efficiency  curve  may  be  assumed  to  hold  for  any  individual  up  to  a  certain  point 
(X),  beyond  which  some  increase  in  yield  is  possible  but  at  an  extraordinary  expense 
of  nutrients  and  other  costs,  illustrated  diagramatically  by  the  broken  curves.  From 
the  standpoint  of  economical  milk  production  it  is  the  point  X  on  the  yield  curve 
that  is  of  primary  importance.  The  advanced  registry  record  as  a  rule  represents  a 
point  sqmewhere  on  the  broken  curve,  often  far  past  the  point  X.  The  broken  curves 
are  highly  speculative. 


developed  into  a  state  where  the  significance  of  the  record  from  the 
standpoint  of  efficiency  of  production  is  very  uncertain. 

Fig.  30  presents  a  more  or  less  diagramatic  attempt  to  compare 
the  advanced  registry  record  with  the  ordinary  record.  The  commercial 


1927] 


PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows 


419 


dairy-man  has  found  by  experience  that  the  best  practice  is  to  breed  his 
cows  to  freshen  about  once  a  year,  regardless  of  the  value  of  the  calf 
at  birth.  The  reason  for  this  is  to  be  found  in  the  great  natural  stimulus 
of  milk  secretion  associated  with  the  reproductive  process,  and  which 


£  4  6  & 

Months  after  Calving 


la 


FIG.  30.  —  COMPARISON  OP  ADVANCED  REGISTRY  RECORD  WITH  ORDINARY  RECORD 
The  A.R.  curve  is  that  of  the  present  Guernsey  records.  As  compared  with  the 
A.R.  record,  the  ordinary  record  starts  at  a  lower  initial  rate  and  declines  at  a  more 
rapid  rate.  Furthermore,  the  cow  is  bred  to  freshen  about  once  a  year  and  allowed 
a  dry  or  rest  period  before  freshening.  Frequent  reproduction  is  essential  to  economi- 
Pal  milk  production,  regardless  of  the  value  of  the  calf.  It  is  the  area  under  the  lower 
type  of  curve  on  which  the  comparison  of  Fig.  29  is  based.  Yield  for  a  year  obtained 
from  the  upper  type  of  curve  cannot  be  used  in  the  comparison  of  Fig.  29. 

becomes  apparent  immediately  following  parturition.  Practical  ex- 
perience has  shown  also  that  it  is  desirable  to  allow  one  to  two  months 
dry  period  before  calving. 

The  importance  of  frequent  reproduction  may  be  demonstrated 
theoretically  on  the  basis  of  the  lactation  curve.  Assume*  a  dry  period 
of  about  six  weeks  and  a  lactation  yield  in  accordance  with  the  general 
equation  for  a  time  two  months  less  than  the  calving  interval.  Let  c 
represent  the  calving  interval  in  months,  then  the  average  yield  per 


month  for  the  interval  is  given  by  a 


It  is  the  average  yield 


over  the  calving  interval  that  is  economically  important,  and  this  may 
be  computed  in  terms  of  a  by  assigning  values  to  A;  and  c.  Table  30  gives 
certain  results  computed  by  this  formula.  Graphic  presentation  is  given 
in  Fig.  31. 

It  is  apparent  from  Table  30  and  Fig.  31  that,  considering  a  per- 
sistency value  of  k  X  103  =  100,  which  is  about  what  may  be  expected 

"Since  this  paper  was  prepared  the  basis  of  the  present  assumption  has  been 
presented  in  more  detail  in  a  paper  submitted  by  the  author  to  the  Journal  of  Dairy 
Science,  entitled  "Milk  yield  in  relation  to  the  recurrence  of  conception". 


420 


BULLETIN  No.  288 


[April, 


under  ordinary  conditions,  the  average  yield  per  month  or  per  year 
decreases  quite  rapidly  as  the  calving  interval  increases.  For  example, 
consider  a  two-year  period  and  an  initial  rate  of  yield  of  1,000  pounds  per 
month,  cow  A  calving  at  the  end  of  12  months  and  again  at  24  months, 


TABLE  30. — AVERAGE  YIELD  PER  MONTH  FOR  THE  CALVING  INTERVAL  COMPUTED 
FROM  THE  LACTATION  CURVE  EQUATIONS 

(Explanation  in  text) 


Calving 
interval 
(months) 

Average  yield  per  month  expressed  as  a  percentage  of 
the  initial  rate  of  yield  per  month 

k  X  103  =  25 

A;  X  103  =  50 

k  X  103  =  100 

k  X  103  =  150 

10  

72.51 
73.27 
73.73 
73.98 
74.05 
73.83 
73.26 
72.47 
71.54 
70.51 

65.94 
65.89 
65.58 
65.08 
64.46 
62.93 
61.19 
59.34 
57.47 
55.59 

55.07 
53.95 
52.68 
51.32 
49.91 
47.09 
44.34 
41.74 
39.30 
37.05 

46.59 
44.89 
43.16 
41.43 
39.75 
36.56 
33.68 
31.09 
28.79 
26.75 

11  

12  

13  

14  

16  

18  

20  

22  

24  

70 


§40 


i* 

o 


®50 


8.2 


20. 


10 


14  \{,  18  ZO 

Calving  Interval  -  Months 


FIG.  31. — SHOWING  THE  NECESSITY  OF  FREQUENT  REPRODUCTION  FOR 
HIGH  AVERAGE  YIELD 

It  is  assumed  that  milk  is  produced  in  accordance  with  the  equation 
for  a  period  two  months  less  than  the  calving  interval.  The  average  yield, 
including  both  lactation  and  dry  period,  is  greater  the  more  frequent  the 
breeding  where  k  X  103  is  not  less  than  50. 


1927]  PERSISTENCY  OP  LACTATION  IN  DAIRY  Cows  421 

cow  B  not  freshening  until  24  months.  The  two  years'  yield  for  A  would 
be  12,643  [  =  2(.5268  X  1000  X  12)]  pounds  while  for  B  it  would  be 
only  8,892  (=  .3705  X  1000  X  24)  pounds.  Furthermore,  until  the  age 
of  maturity  the  initial  rate  of  yield  ordinarily  increases  with  each  calf, 
which  would  give  a  still  further  advantage  to  cow  A,  breeding  every  12 
months.  The  illustration  serves  to  show  the  great  influence  of  frequent 
freshening  on  the  average  yearly  yield  over  a  series  of  years.  The  glamour 
of  a  large  year-record  representing  only  the  first  365  days  of  a  much 
longer  lactation  period  should  not  obscure  the  economic  issue. 

In  the  economically  practical  type  of  lactation  curve  (ordinary 
record  of  Fig.  30),  we  have  a  lactation  yield  equal  to,  say,  ten  months' 
yield  by  the  equation,  and  with  a  persistency  value  of  about  k  X  103  = 
100.  It  becomes  of  interest  to  determine  the  point  in  the  lactation 
curve  at  which  the  ten  months'  yield  is  unaffected  by  variability  in  per- 
sistency. The  solution  is  given  by  the  value  of  i'  in  the  equation  t'  = 

I        I0e~l0k 

-.     For  the  value  of  k  X  103  =  100,  t'  =  4.18.     That  is, 
k      1  —  e-lok 

under  the  conditions  of  commercial  practice  a  short-time  test  should 
afford  the  best  measure  of  the  cow's  production  if  conducted  at  4.2 
months  after  calving. 

The  chief  objection  put  forward  to  the  short-time  test  is  that  it 
Tdoes  not  measure  persistency  of  lactation  and  consequently  does  not 
accurately  indicate  yearly  production.  As  a  matter  of  fact,  a  short-time 
test  at  the  beginning  of  lactation  is  a  better  measure  of  persistency 
than  is  the  yield  for  a  year.  This  statement  is  warranted  by  the  fact  that 
the  correlation  between  theoretical  initial  rate  of  yield  and  persistency 
(rate  of  decrease)  for  the  present  Guernsey  records  is  r  =  .535,  whereas 
the  correlation  between  yield  for  the  year  and  persistency  is  only  r  = 
-.226.  Yield  for  the  year  of  course  embodies  the  result  of  persistency, 
but  any  accurate  measure  of  this  character  must  be  based  directly  on 
the  lactation  curve. 

It  is  clear  that  if  the  short-time  test  is  conducted  at  the  stage  of 
lactation  where  the  yield  for  the  portion  of  lactation  under  consider- 
ation is  independent  of  variability  in  persistency,  it  becomes  theoretically 
a  perfect  index  of  the  yield  for  the  longer  period.  For  the  purpose  of 
representing  the  yearly  production  of  the  cow  under  ordinary  condi- 
tions, bearing  a  calf  yearly,  it  would  seem  that  a  short-time  test  con- 
ducted during  the  fourth  month  of  lactation  should  afford  a  highly 
valuable  measure  of  the  cow's  production  from  a  practical  standpoint. 
The  yield  for  such  a  short-time  test,  like  the  yield  for  the  year*  would 
embody  the  result  of  persistency  but  would  give  no  accurate  measure 
of  persistency. 


422  BULLETIN  No.  288  [April, 

As  compared  with  initial  rate  of  yield,  or  rate  of  yield  at  later  stages 
of  lactation,  persistency  is  a  much  less  important  factor  in  determining 
the  ordinary  lactation  yield.  Differences  in  persistency  are  evidently 
very  much  subject  to  differences  in  factors  of  an  environmental  nature, 
and  very  little  subject  to  differences  in  factors  of  a  genetic  nature,  so 
far  as  indicated  by  the  present  records.  From  the  inheritance  stand- 
point, however,  the  present  data  are  not  extensive  enough  to  establish 
satisfactorily  the  true  condition  of  affairs,  and  judgment  should  be 
reserved  pending  investigation  of  a  larger  number  of  records. 

On  the  other  hand,  as  compared  with  persistency,  initial  rate  of 
yield  is  a  much  more  powerful  factor  determining  the  ordinary  lactation 
yield;  it  is  less  subject  to  environmental  factors;  and  it  seems  to  promise 
considerable  opportunity  of  improvement  by  selective  breeding.  Cows 
with  inherent  capacity  for  high  yearly  yield  are  clearly  necessary  to  the 
most  efficient  production  of  milk.  High  yield  for  the  year  is  to  be  at- 
tained only  thru  high  initial  rate  of  yield.  Note,  in  Table  12,  how  dis- 
tinctly the  correlation  surface  of  initial  rate  of  yield  and  yield  for  the 
year  is  cut  off  at  the  upper  right  border. 

It  is  a  question  for  serious  consideration  whether  we  may  not 
progress  as  well  or  better  in  breeding  and  selecting  high-yielding  cows 
on  the  basis  of  a  short-time  yield  soon  after  calving,  as  we  may  on  the 
basis  of  a  short-time  test  later  in  the  lactation,  or  on  the  basis  of  the 
lactation  yield  itself.  Given  high  initial  rate  of  yield,  and  regular,  fre- 
quent reproduction,  persistency  of  lactation  seems  to  be  of  minor  im- 
portance in  the  problem  of  breeding  and  selecting  efficient  cows. 


SUMMARY  AND  CONCLUSIONS 

dy 
A  curve  of  the  type  -j-  =  Ae~kt  has  been  fitted  to  each  of  1,534 

Guernsey  records  and  1,395  Holstein  records.  In  the  equation  y  is  yield, 

dii 

-5-  is  the  rate  of  yield,  A  is  the  theoretical  initial  rate  of  yield,  t  is  time 

from  calving,  and  k  is  the  rate  of  decrease  in  the  rate  of  yield.  The  value 
of  k  is  used  as  a  measure  of  persistency  of  lactation.  The  value  of  A 
is  representative  of  the  rate  of  yield  shortly  after  calving.  The  A  and  k 
constants  of  the  individual  curves  have  been  studied  statistically,  and  in 
case  of  the  Guernsey  records,  the  yield  for  the  year  and  fat  percentage 
for  the  year,  also.  Yield  has  been  measured  on  an  energy  basis  in  terms 
of  4-percent  milk. 

It  was  found  that  A  and  k  are  quite  closely  related :  for  the  Guernsey 
records  r  =  .535;  for  the  Holstein  records  r  =  .433.    In  both  breeds  A 


1937]  PERSISTENCY  OF  LACTATION  IN  DAIRY  Cows  423 

increases  with  age  to  about  nine  years  and  then  declines  while  fc  is 
independent  of  age  with  A  constant. 

Yearly  yield  is  more  closely  related  to  A  than  to  k:  r  =  .672  and 
r  =  -.226  respectively.  The  correlation  between  yearly  yield  and  rate 
of  yield  increases  with  advance  in  lactation  up  to  six  months,  after  which 
it  decreases.  At  six  months  r  =  .980  for  the  smoothed  lactation  curve 
and  r  =  .935  for  the  raw  lactation  curve.  From  the  mathematical 
properties  of  the  lactation  curve,  it  seems  that  for  the  ordinary  lactation 
of  ten  months  and  k  =  .1,  the  best  time  of  conducting  a  short-time  test 
to  estimate  the  lactation  yield  is  4.2  months  after  calving. 

The  corrected  persistency  values  show  a  high  degree  of  variability, 
the  coefficient  of  variability  for  the  Guernsey  and  Holstein  records  being 
60.97  and  46.25  respectively;  and  the  corresponding  coefficients  for 
yearly  yield  as  affected  by  persistency,  14.53  and  13.05. 

The  mean  initial  rate  corrected  to  age  of  maximum  is,  for  the 
Guernsey  records,  47.2;  and  for  the  Holstein  records,  81.6  pounds 
4-percent  milk  per  day.  Corrected  to  the  same  initial  rate,  the  Holstein 
records  show  greater  persistency  (lower  value  of  k)  than  the  Guernsey 
records. 

Environment  has  a  great  effect  on  persistency.  The  evidence  as 
to  the  influence  of  heredity  on  persistency  with  environment  constant 
is  inconclusive  and  conflicting :  half-sisters  by  Guernsey  sires  show  no 
correlation  r  =  —  .001  +  .029;  half-sisters  by  Holstein  sires,  a  feeble 
correlation,  r  =  .111  ±  .016.  Half-sisters  by  a  common  dam  show  a 
material  correlation,  r  =  .418  ±  .080  (Guernsey)  and  r  =  .390  ±  .065 
(Holstein).  Dam-and-daughter  relationships  show  r  =  .259  ±  .086 
(Guernsey)  and  r  =  .370  ±  .059  (Holstein).  Full  sisters  in  both  breeds 
show  a  non-significant  negative  correlation,  r  =  —  .178  ±  .106  (Guern- 
sey) and  r  =  —  .090  +  .084  (Holstein).  Successive  lactations  of  the 
same  cow  (Holstein)  show  a  correlation  of  r  =  .370  +  .059.  All  the 
correlations  of  this  paragraph  are  for  persistency  corrected  for  initial 
rate  of  yield,  and  the  correlations  are  corrected  to  eliminate  the  effect 
of  common  environment. 

In  view  of  the  uncertainty  of  effecting  further  genetic  improvement 
in  our  dairy  breeds  with  respect  to  persistency  of  lactation,  the  low  cor- 
relation between  persistency  and  yearly  yield,  and  the  high  correlation 
between  rate  of  yield  and  yearly  yield,  it  must  be  granted  that  a  properly 
conducted  short-time  test  may  serve  as  an  excellent  production  record, 
from  both  economical  and  biological  standpoints. 


424  BULLETIN  No.  288 


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